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Bipartite Domination and Simultaneous Matroid Covers

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Abstract

Damaschke, Müller, and Kratsch [Inform. Process. Lett., 36 (1990), pp. 231--236] gave a polynomial-time algorithm to solve the minimum dominating set problem in convex bipartite graphs B=(XY,E)B=(X \cup Y,E), that is, where the nodes in Y can be ordered so that each node of X is adjacent to a contiguous sequence of nodes. Gamble et al. [Graphs Combin., 11 (1995), pp. 121--129] gave an extension of their algorithm to weighted dominating sets. We formulate the dominating set problem as that of finding a minimum weight subset of elements of a graphic matroid, which covers each fundamental circuit and fundamental cut with respect to some spanning tree T. When T is a directed path, this simultaneous covering problem coincides with the dominating set problem for the previously studied class of convex bipartite graphs. We describe a polynomial-time algorithm for the more general problem of simultaneous covering in the case when T is an arborescence. We also give NP-completeness results for fairly specialized classes of the simultaneous cover problem. These are based on connections between the domination and induced matching problems.

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... They introduced and defined the concept of bipartite dominating set and bipartite domination number, which motivates this study. Their concept of the bipartite domination was inspired by the study of Ko, C., and Shepherd, F., [2]. In line with this, they investigated the case that a dominating set must induce a bipartite subgraph. ...
... Conversely, suppose S satisfies property (1). Then, clearly S is a bipartite dominating set in G ∨ H. Now, suppose S satisfies property (2). Since for each u, v ∈ B1 ⊆ V (G), u / ∈ NG(v) and v / ∈ NG(u), it follows that S is a dominating set and G ∨ H[S] is a bipartite graph. ...
... Conversely, suppose S satisfies property (1). Then, clearly S is a bipartite dominating set in G ∨ H. Similarly for S satisfying property (2). Now, suppose S satisfies property (3). ...
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For a nontrivial connected graph G, a non-empty set S ⊆ V (G) is a bipartite dominating set of graph G, if the subgraph G[S] induced by S is bipartite and for every vertex not in S is dominated by any vertex in S. The bipartite domination number denoted by γ bip (G) of graph G is the minimum cardinality of a bipartite dominating set G. In this paper, we determine the exact bipartite domination number of path graph and cycle graph via congruence modulo. Moreover, this study generates the possible exact values of the bipartite domination number of the complete graph, complete bipartite graph, join graph, fan graph and wheel graph.
... We need propositions 2.5 -2.7 to prove proposition 2. 9. But before giving this proposition we will present proposition 2.8 in order to show that if G has a node Chapter 2. Related work for which none of three conditions cited in propositions 2.5 -2.7 is satisfied, then inequality 2. 16 is not a facet of P (G, b) ...
... Then the inequality 2. 16 does not define a facet of P (G, b). ...
... According to Figure 3.6, these 3 intervals cover the interval [c 1 , c 10 ].To reach y = c 16 , it is clear that we should at least cover the interval [c 10 , c 16 ] using intervals starting after c 4 (because the third interval starts at c 3 ). Since we already used 3 intervals to reach c 10 , we should use at most k − 3 = 3 intervals to reach y = c16 . Intervals numbered from 4 to 6 necessarily constitute an optimal solution of the problem that consists in covering [c 10 , c 16 ] by no more than 3 intervals starting after c 4 . ...
Thesis
Let G be a vertex-weighted undirected graph. We aim to compute a minimum weight subset of vertices whose removal leads to a graph where the size of each connected component is less than or equal to a given positive number k. If k = 1 we get the classical vertex cover problem. Many formulations are proposed for the problem. The linear relaxations of these formulations are theoretically compared. A polyhedral study is proposed (valid inequalities, facets, separation algorithms). It is shown that the problem can be solved in polynomial time for many special cases including the path, the cycle and the tree cases and also for graphs not containing some special induced sub-graphs. Some (k + 1)-approximation algorithms are also exhibited. Most of the algorithms are implemented and compared. The k-separator problem has many applications. If vertex weights are equal to 1, the size of a minimum k-separator can be used to evaluate the robustness of a graph or a network. Another application consists in partitioning a graph/network into different sub-graphs with respect to different criteria. For example, in the context of social networks, many approaches are proposed to detect communities. By solving a minimum k-separator problem, we get different connected components that may represent communities. The k-separator vertices represent persons making connections between communities. The k-separator problem can then be seen as a special partitioning/clustering graph problem
... Given a graph G, the cardinality of a maximum induced matching is denoted by iµ(G). MIM, also NP-hard in general, remains NP-hard in several graph classes such as planar graphs of degree at most 4 [16], line graphs [17], bipartite graphs [23], bipartite graphs of girth at least 14 [2], or of girth at least 6 [28] or of degree at most 3 [18]. This last result has been recently improved in [7] by proving that MIM is NP-hard in k-regular bipartite graphs for any k ≥ 3. On the other hand, MIM can be polynomially solved in weakly chordal graphs [4], AT-free graphs [5,3], circular arc graphs [11], cocomparability graphs [12], graphs of bounded clique-width [17], chordal graphs [2] and interval filament graphs [3], which include cocomparability graphs and polygoncircle graphs, where the latter include circle graphs, circular-arc graphs, chordal graphs, and outerplanar graphs. ...
... MIM is known to be NP-hard in planar graphs of maximum degree 4 [16]. We can even adapt the proof to show a slightly more restrictive result: ...
... Proof: The proof of [16] is based on the remark that, given a graph G and adding for each vertex v a vertex v ′ and a pendant edge vv ′ , the resulting graph G ′ satisfies iµ(G ′ ) = α(G), where α denotes the stability number. Moreover if G is planar then G ′ is also planar. ...
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Graph Theory International audience Given a graph, finding the maximal matching of minimum size (MMM) and the induced matching of maximum size (MIM) have been very popular research topics during the last decades. In this paper, we give new complexity results, namely the NP-hardness of MMM and MIM in induced subgrids and we point out some promising research directions. We also sketch the general framework of a unified approach to show the NP-hardness of some problems in subgrids.
... The concept of induced matching was introduced by Stockmeyer and Vazirani as the "risk-free" marriage problem in 1982 [45]. Since then, this concept, and the corresponding Induced Matching problem, have been studied extensively due to their wide range of applications and connections to other graph problems [11,13,29,32,38,45]. Similarly, the Acyclic Matching problem considers a graph G and a positive integer ℓ, and asks whether G contains an acyclic matching of size at least ℓ. ...
... The Induced Matching problem exhibits different computational complexities depending on the class of graphs considered. It is known to be NP-complete for bipartite graphs of maximum degree 4 [45], k-regular graphs for k ≥ 4 [48], and planar graphs of maximum degree 4 [32]. On the positive side, the problem is known to be polynomial-time solvable for many classes of graphs, such as chordal graphs [11], chordal bipartite graphs [12], trapezoid graphs, interval-dimension graphs, and cocomparability graphs [26]. ...
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A \emph{matching} is a subset of edges in a graph G that do not share an endpoint. A matching M is a \emph{P\mathcal{P}-matching} if the subgraph of G induced by the endpoints of the edges of M satisfies property P\mathcal{P}. For example, if the property P\mathcal{P} is that of being a matching, being acyclic, or being disconnected, then we obtain an \emph{induced matching}, an \emph{acyclic matching}, and a \emph{disconnected matching}, respectively. In this paper, we analyze the problems of the computation of these matchings from the viewpoint of Parameterized Complexity with respect to the parameter \emph{treewidth}.
... By doing so, every pair of vertices in M is already connected to their edges from G. Thus, it forms an induced subgraph of G [Cameron, 1989;Ko and Shepherd, 2003]. Given a positive integer threshold k, a decision version for the MIM problem can be: is it possible to find an induced matching with size |M| ≥ k? ...
... • the graph obtained by the network of pedestrian ways is planar with maximum degree equal to four [Ko and Shepherd, 2003]. ...
Thesis
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Commuting is a routine task of people living in urban areas. Shared mobility services aim to offer different options to this routine by providing better comfort and faster trips than conventional public transport means, along with avoiding the clients’ costs of owning a private vehicle. A properly planned carsharing service can be attractive even for who owns and drives a private vehicle but would consider not owning it anymore if a cheaper and more sustainable transport mean is available. Low-cost carsharing rentals can be achieved by suitably positioning the fleet along the city and by making the most of the shared vehicles according with a previous selection of which subset of trip demands can be served. This previous selection would choose which demands have a combining origin and destination, allowing these clients to use a same vehicle but in different moments, not requiring the carsharing company to relocate the fleet among stations due to different demands along the day and week. This work contextualizes the operational and computational challenges in planning a carsharing service; proves the NP-Completeness of optimizing the locations for shared mobility stations; proposes a Mixed-Integer Linear Programming formulation for this original problem, and another Mixed-Integer Linear Programming formulation which yields good locations for stations; and applies a polynomial time linear formulation to simulate and compare the performance of three different carsharing business models according with historical mobility data from the São Paulo Metropolitan Area. Results show that it is possible to offer a profitable low-cost carsharing service without performing vehicle relocations. However, only a subset of trips are served and clients must be flexible enough to walk to get to an available vehicle nearby. Results also demonstrate that trips selected to be served are similar among the different business models; are concentrated on São Paulo’s downtown region; are shorter than the average trip, but otherwise behave in a similar way as compared to the complete set of trips; and the lack of parking slots may be a risk to the carsharing company.
... First Stockmeyer and Vazirani [16] and Cameron [3] proved that it is NP-hard to compute Σ for bipartite graphs. This result was essentially improved by Ko and Shepherd [8]. Theorem 2 (Ko and Shepherd [8]) For each k ≥ 1, Σ is NP-hard to compute for the class of planar, bipartite graphs of maximum degree 4 with one side of the bipartition consisting only of degree two vertices and with each cycle having length ≡ 0 (mod 2 k ). ...
... This result was essentially improved by Ko and Shepherd [8]. Theorem 2 (Ko and Shepherd [8]) For each k ≥ 1, Σ is NP-hard to compute for the class of planar, bipartite graphs of maximum degree 4 with one side of the bipartition consisting only of degree two vertices and with each cycle having length ≡ 0 (mod 2 k ). Recall that a dominating set D is a subset of V (G) such that each vertex in V (G) \ D is adjacent to a vertex of D. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set. ...
... Recently, these problems attracted much attention because of their theoretical interest and practical motivation -see the papers mentioned above and also [4,[12][13][14][15]25,36,34,38,41,49]. Stockmeyer and Vazirani [46] have shown that for every k ≥ 2, the MDkM Problem is NP-complete even for bipartite graphs of maximum degree 4. In particular, the MIM Problem is NP-complete on bipartite graphs which was shown independently by Cameron [12]. ...
... Stockmeyer and Vazirani [46] have shown that for every k ≥ 2, the MDkM Problem is NP-complete even for bipartite graphs of maximum degree 4. In particular, the MIM Problem is NP-complete on bipartite graphs which was shown independently by Cameron [12]. It is NP-complete even on planar bipartite graphs of maximum degree 4 as shown in [36]. The MIM problem remains NP-complete for line graphs [34] and thus also for claw-free graphs. ...
Conference Paper
For a finite undirected graph G = (V,E) and positive integer k ≥ 1, an edge set M ⊆ E is a distance-k matching if the mutual distance of edges in M is at least k in G. For k = 1, this gives the usual notion of matching in graphs, and for general k ≥ 1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k = 2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers. Finding a maximum induced matching is NP\mathbb{NP}-complete even on very restricted bipartite graphs but for k = 2, it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)2 of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G. We show that, unlike for k = 2, for a chordal graph G, L(G)3 is not necessarily chordal, and finding a maximum distance-3 matching remains NP\mathbb{NP}-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-3 matching problem can be solved in polynomial time. Moreover, we obtain various new results for induced matchings.
... In many graph classes such as trees [6], chordal graphs [2], circular-arc graphs [5], and interval graphs [6], a maximum induced matching can be found in polynomial time. However, Maximum Induced Matching is NP-hard in planar 3-regular graphs or planar bipartite graphs with degree-2 vertices in one part and degree-3 vertices in the other part [4,10,17]. Kobler and Rotics [11] proved the NP-hardness of this problem in Hamiltonian graphs, claw-free graphs, chair-free graphs, line graphs, and regular graphs. The applications of induced matchings are diverse and include secure communication channels, VLSI design, and network flow problems, as demonstrated by Golumbic and Lewenstein [6]. ...
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An induced subgraph is called an induced matching if each vertex is a degree-1 vertex in the subgraph. The \textsc{Almost Induced Matching} problem asks whether we can delete at most k vertices from the input graph such that the remaining graph is an induced matching. This paper studies parameterized algorithms for this problem by taking the size k of the deletion set as the parameter. First, we prove a 6k-vertex kernel for this problem, improving the previous result of 7k. Second, we give an O(1.6957k)O^*(1.6957^k)-time and polynomial-space algorithm, improving the previous running-time bound of O(1.7485k)O^*(1.7485^k).
... Stockmeyer and Vazirani introduced the concept of induced matching as the "risk-free" marriage problem in 1982 [41]. Since then, induced matchings have been studied extensively due to its various applications and relations with other graph problems [8,10,15,27,28,29,31,36,39,41,44]. Given a graph G and a positive integer ℓ, Induced Matching Below Triviality (IMBT) asks whether G has an induced matching of size at least ℓ with the parameter k = n 2 − ℓ, where Here, c 4 and tw denote the number of cycles with length four and the treewidth of the input graph, respectively. ...
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A matching M in a graph G is an \emph{acyclic matching} if the subgraph of G induced by the endpoints of the edges of M is a forest. Given a graph G and a positive integer \ell, Acyclic Matching asks whether G has an acyclic matching of size (i.e., the number of edges) at least \ell. In this paper, we first prove that assuming W[1]FPT\mathsf{W[1]\nsubseteq FPT}, there does not exist any FPT\mathsf{FPT}-approximation algorithm for Acyclic Matching that approximates it within a constant factor when the parameter is the size of the matching. Our reduction is general in the sense that it also asserts FPT\mathsf{FPT}-inapproximability for Induced Matching and Uniquely Restricted Matching as well. We also consider three below-guarantee parameters for Acyclic Matching, viz. n2\frac{n}{2}-\ell, MM(G)\mathsf{MM(G)}-\ell, and IS(G)\mathsf{IS(G)}-\ell, where n is the number of vertices in G, MM(G)\mathsf{MM(G)} is the matching number of G, and IS(G)\mathsf{IS(G)} is the independence number of G. Furthermore, we show that Acyclic Matching does not exhibit a polynomial kernel with respect to vertex cover number (or vertex deletion distance to clique) plus the size of the matching unless \mathsf{NP}\subseteq\mathsf{coNP}\slash\mathsf{poly}.
... They investigated the case that S must induce a bipartite subgraph. Their idea about the bipartite domination came from the published study entitled "Bipartite domination and simultaneous matroid covers" [2]. ...
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For a nontrivial connected graph G, a non-empty set S ⊆ V (G) is a bipartite dominating set of graph G, if the subgraph G[S] induced by S is bipartite and for every vertex not in S is dominated by any vertex in S. The bipartite domination number denoted by γ bip (G) of graph G is the minimum cardinality of a bipartite dominating set G. In this paper, we determine the exact bipartite domination number of a crown graph and its mycielski graph as well as the bipartite domination number of the mycielski graph of path and cycle graphs.
... Then we de ne the bipartite domination number ( ) of graph as the minimum size of a bipartite dominating set. We note that the term "bipartite domination" has also been used to describe the domination problem in bipartite graphs, such as in [11]. On the other hand, it is used in our way in [9]. ...
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The bipartite domination number of a graph is the minimum size of a dominating set that induces a bipartite subgraph. In this paper we initiate the study of this parameter, especially bounds involving the order, the ordinary domination number, and the chromatic number. For example, we show for an isolate-free graph that the bipartite domination number equals the domination number if the graph has maximum degree at most 3; and is at most half the order if the graph is regular, 4-colorable, or has maximum degree at most 5.
... It is known that solving MIM on special instances such as trees and interval graphs [11], chordal graphs [1], circular arc graphs [10] etc. can be done in polynomial time. On the other hand, it is also known to be NP-hard in bipartite graphs with maximum degree 4, planar 3-regular graphs, planar bipartite graphs with degree 2 vertices on one component and degree 3 vertices in the other component, Hamiltonian graphs, claw-free graphs, line graphs and regular graphs [4,13,14]. ...
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The Maximum Induced Matching problem asks to find the maximum k such that, given a graph G=(V,E), can we find a subset of vertices S of size k for which every vertices v in the induced graph G[S] has exactly degree 1. In this paper, we design an exact algorithm running in O(1.2630n)O(1.2630^n) time and polynomial space to solve the Maximum Induced Matching problem for graphs where each vertex has degree at most 3. Prior work solved the problem by finding the Maximum Independent Set using polynomial space in the line graph L(G2)L(G^2); this method uses O(1.3139n)O(1.3139^n) time.
... The Induced Matching problem was first introduced by Stockmeyer and Vazirani [29] as a variant of the maximum matching problem and proved to be NP-complete in general graphs. This problem is known to be NP-complete for planar graphs of maximum degree 4 [18], bipartite graphs of maximum degree three, C 4 -free bipartite graphs [30], r−regular graphs for r ≥ 5, line-graphs, chair-free graphs, and Hamiltonian graphs [19]. The problem is known to be polynomial time solvable for many classes of graphs such as trees [31], chordal graphs [5] and line-graphs of Hamiltonian graphs [19] (for more information see e.g. ...
Preprint
A matching is a set of edges in a graph with no common endpoint. A matching M is called acyclic if the induced subgraph on the endpoints of the edges in M is acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for an acyclic matching of size k in G. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs with maximum degree three and girth of arbitrary large. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter k. On the other hand, the problem is fixed parameter tractable with respect to k, for line graphs, C4C_4-free graphs and every proper minor-closed class of graphs (including bounded tree-width and planar graphs).
... A maximum induced matching can be found in polynomial time in many graph classes, such as trees [7], chordal graphs [2], circular-arc graphs [8] and interval graphs [7]. However, Maximum Induced Matching is NP-hard even in planar 3-regular graphs or planar bipartite graphs with degree-2 vertices in one part and degree-3 vertices in the other part [4,11,19]. The NP-hardness of this problem in Hamiltonian graphs, claw-free graphs, chair-free graphs, line graphs, and regular graphs was proved by Kobler and Rotics [12]. ...
Article
The Almost Induced Matching problem asks whether we can delete at most k vertices from the input graph such that each vertex in the remaining graph has a degree exactly one. This paper studies parameterized algorithms for this problem by taking the size k of the deletion set as the parameter. We give a 7k-vertex kernel and an O⁎(1.7485k)-time and polynomial-space algorithm, both of which are the best-known results. The linear-vertex kernel is obtained by using an extended crown decomposition and careful analysis, and the parameterized algorithm is based on a branch-and-search paradigm.
... The subdivision SU B(G) of a graph G is a bipartite graph and it has ι(G) = γ(G) [17]. Moreover, Ko and Shepherd [10] proved that ...
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Given a graph G, the P3-convex hull (resp. P3⁎-convex hull) of a set C ⊆ V (G) is obtained by iteratively adding to C vertices with at least two neighbors inside C (resp. at least two non-adjacent neighbors inside C). A P3-Helly-independent (resp. P3⁎-Helly-independent) of a graph G is a set S ⊆ V (G) such that the intersection of the P3-convex hulls (P3⁎-convex hulls) of S \ {v} (∀v ∈ S) is empty. We denote by P3-Helly number (resp. P3⁎-Helly number) the size of a maximum P3-Helly-independent (resp. P3⁎-Helly-independent). The edge counterparts of these two P3-Helly-independents follow the same restrictions applied to its edges. The vp3hi (resp. vsp3hi, ep3hi, and esp3hi) problem aims to determine the P3-Helly number (resp. P3⁎-Helly number, edge P3-Helly number, and edge P3⁎-Helly number) of a graph. We establish the computational complexities of vp3hi, vsp3hi, ep3hi, and esp3hi for a collection of graph classes, including bipartite graphs, split graphs, and join of graphs.
... However, it is known to be NP-hard even for very restricted graph classes. For instance, it is known that MIM remains NP-hard in planar 3-regular graphs [11], in planar bipartite graphs with degree 2 vertices in one part and degree 3 vertices in the other part [23], in k-regular bipartite graphs for any k ≥ 3 [8], and in Hamilto- nian graphs, claw-free graphs and line graphs [24]. On the other hand, MIM is polynomial-time solvable in trees [19], chordal graphs [5], circular arc graphs [18] and interval graphs [19]. ...
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A graph G is called well-indumatched if all of its maximal induced matchings have the same size. In this paper we characterize all well-indumatched trees. We provide a linear time algorithm to decide if a tree is well-indumatched or not. Then, we characterize minimal well-indumatched graphs of girth at least 9 and show subsequently that there is no well-indumatched graph of odd girth g greater than or equal to 9 and different from 11. On the other hand, there are infinitely many well-indumatched unicyclic graphs of girth k, where k is in {3, 5, 7} or k is an even integer greater than 2. We also show that, although the recognition of well-indumatched graphs is known to be co-NP-complete in general, one can recognize in polynomial time well-indumatched graphs where the size of maximal induced matchings is fixed.
... Although MIM is polynomial-time solvable in trees [9], chordal graphs [2], circular arc graphs [8], interval graphs [9] and many others, it has been known to be NP-hard in bipartite graphs with maximum degree 4 for more than 30 years [18]. In fact, it remains NP-hard even in planar 3-regular graphs or in planar bipartite graphs with degree-2 vertices in one part and degree-3 vertices in the other part [4,12]. Kobler and Rotics [13] also showed the NP-hardness of MIM in Hamiltonian graphs, claw-free graphs, chair-free graphs, line graphs and regular graphs. ...
Article
This paper studies exact algorithms for the Maximum Induced Matching problem, in which an n-vertex graph is given and we are asked to find a set of maximum number of edges in the graph such that no pair of edges in the set have a common endpoint or are adjacent by another edge. This problem has applications in many different areas. We give several structural properties of the problem and show that the problem can be solved in O*(1.4231n) time and polynomial space or O*(1.3752n) time and exponential space.
... Induced matchings have applications in communication network testing [47], concurrent transmission of messages in wireless ad hoc networks [2] and secure communication channels in broadcast networks [30]. The Maximum Induced Matching problem was also studied in [8,9,10,11,12,13,14,19,22,38,39,42,43], see [10,22] also for a survey and new results. In [39], Kobler and Rotics showed that the graphs where the sizes of a maximum matching and a maximum induced matching coincide, can be recognized in polynomial time. ...
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A graph is . well-indumatched if all its maximal induced matchings are of the same size. We first prove that recognizing whether a graph is well-indumatched is a co-NP-complete problem even for . (2P5,K1,5)-free graphs. We then show that decision problems . Independent Dominating Set, . Independent Set, and . Dominating Set are NP-complete for the class of well-indumatched graphs. We also show that this class is a co-indumatching hereditary class, i.e., it is closed under deleting the end-vertices of an induced matching along with their neighborhoods, and we characterize well-indumatched graphs in terms of forbidden co-indumatching subgraphs. We prove that recognizing a co-indumatching subgraph is an NP-complete problem. We introduce a . perfectly well-indumatched graph, in which every induced subgraph is well-indumatched, and characterize the class of these graphs in terms of forbidden induced subgraphs. Finally, we show that the weighted versions of problems . Independent Dominating Set and . Independent Set can be solved in polynomial time for perfectly well-indumatched graphs, but problem . Dominating Set is NP-complete.
... A maximum induced matching can be found in polynomial time in many graph classes, such as trees [7], chordal graphs [2], circular arc graphs [8] and interval graphs [7]. However, Maximum Induced Matching is NP-hard even in planar 3-regular graphs or planar bipartite graphs with degree-2 vertices in one part and degree-3 vertices in the other part [4,11,19]. The NP-hardness of this problem in Hamiltonian graphs, claw-free graphs, chair-free graphs, line graphs and regular graphs is proved by Kobler and Rotics [12]. ...
Conference Paper
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The Almost Induced Matching problem asks whether we can delete at most k vertices from a graph such that the remaining graph is an induced matching, i.e., a graph with each vertex of degree 1. This paper studies parameterized algorithms for this problem by taking the size of deletion set k as the parameter. By using the techniques of finding maximal 3-path packings and an extended crown decomposition, we obtain the first linear vertex kernel for this problem, improving the previous quadratic kernel. We also present an O * (1.7485 k)-time and polynomial-space algorithm, which is the best known parameterized algorithm for this problem.
... Although MIM is polynomial-time solvable in trees [9], chordal graphs [2], circular arc graphs [8], interval graphs [9] and many others, it has been known to be NP-hard in bipartite graphs with maximum degree 4 for more than 30 years [15]. In fact, it remains NP-hard even in planar 3regular graphs or in planar bipartite graphs with degree-2 vertices in one part and degree-3 vertices in the other part [4,11]. Kobler and Rotics [12] also showed the NP-hardness of MIM in Hamiltonian graphs, claw-free graphs, chair-free graphs, line graphs and regular graphs. ...
Conference Paper
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This paper studies exact algorithms for the Maximum Induced Matching problem, in which an n-vertex graph is given and we are asked to find a set of maximum number of edges in the graph such that no pair of edges in the set have a common endpoint or are adjacent by another edge. This problem has applications in many different areas. We will give several structural properties of the problem and present an O(1.4391n)O^*(1.4391^n)-time algorithm, which improves previous exact algorithms for this problem.
... The Maximum Induced Matching problem is NP-complete for bipartite graphs [9,55] and bipartite graphs with a maximum vertex degree of 3 [39], C 4 -free bipartite graphs [39], line graphs [37] and for planar graphs with a maximum vertex degree of 4 [36], but on the other hand, it is polynomially solvable for chordal [9] and weakly chordal graphs [12], circular-arc graphs [24], AT-free graphs [10,14], (P k , K 1,n )-free graphs (for any positive k and n) [42], and graphs where a maximum matching and a maximum induced matching have the same size [13,37]. Regarding polynomial-time approximability, it is known that the Maximum Induced Matching problem is APX-complete on r-regular graphs for all r 3, and bipartite graphs with a maximum vertex degree of 3 [18]. ...
... The Maximum Induced Matching problem is NP-complete for bipartite graphs [8,26,40], line graphs [25] and for planar graphs with maximum vertex degree of 4 [24], but on the other hand, it is polynomially solvable for chordal [8] and weakly chordal graphs [11], circular-arc graphs [16] and AT-free graphs [9,12]. ...
... We prove NP-completeness for the problem with σ(G) even if graphs have maximal induced matchings of at most two sizes. Ko and Shepherd [19] investigated relations between Σ(G) and γ(G), the domination number of G. They mentioned that they do not know any class of graphs for which exactly one of γ, Σ is polynomial-time computable. ...
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A graph is well-indumatched if all its maximal (with respect to set inclusion) induced matchings are of the same size. We first prove that recognizing the class WIM of well-indumatched graphs is a co-NP-complete problem even for (2P5, K1,5)-free graphs. We then show that the well-known decision problems such as Independent Dominating Set, Independent Set, and Dominating Set are NP-complete for well-indumatched graphs. We also show that WIM is a co-indumatching hereditary class and characterize well-indumatched graphs in terms of forbidden co-indumatching subgraphs. However, we prove that recognizing co-indumatching subgraphs is an NP-complete problem. A graph G is perfectly well-indumatched if every induced subgraph of G is well-indumatched. We characterize the class of perfectly well-indumatched graphs in terms of forbidden induced subgraphs. Finally, we show that both Independent Dominating Set and Independent Set can be solved in polynomial time for perfectly well-indumatched graphs, even in their weighted versions, but Dominating Set is still NP-complete.
... First of all, this problem is NP-hard, which was proved independently in [5,33]. Moreover, it remains NP-hard under substantial restrictions, for instance: for bipartite graphs of vertex degree at most 3 [27,32], line graphs [26], planar graphs of vertex degree at most 4 [25] and even for cubic planar graphs [11]. On the other hand, polynomial-time algorithms for this problem have been developed for weakly chordal graphs [7], AT-free graphs [8], circular arc graphs [19], graphs of bounded clique-width [26] and some other classes of graphs [3,4,6,20,27]. ...
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The maximum induced matching problem is known to be APX-hard in the class of bipartite graphs. Moreover, the problem is also intractable in this class from a parameterized point of view, i.e. it is W[1]-hard. In this paper, we reveal several classes of bipartite (and more general) graphs for which the problem admits fixed-parameter tractable algorithms. We also study the computational complexity of the problem for regular bipartite graphs and prove that the problem remains APX-hard even under this restriction. On the other hand, we show that for hypercubes (a proper subclass of regular bipartite graphs) the problem admits a simple solution.
... We prove NP-completeness for the problem with σ(G) even if graphs have maximal induced matchings of at most two sizes. Ko and Shepherd (2003) investigated relations between Σ(G) and γ(G), the domination number of G. They mentioned that they do not know any class of graphs for which exactly one of γ, Σ is polynomial-time computable. ...
Conference Paper
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... This problem has been intensively studied in recent years. It is known to be NP-complete for planar graphs of maximum degree 4 [17] , bipartite graphs of maximum degree 3, r-regular graphs for r ≥ 5, line-graphs and ...
Conference Paper
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Preprint
An induced matching in a graph is a set of edges whose endpoints induce a 1-regular subgraph. Gupta et al. (2012,\cite{Gupta}) showed that every n-vertex graph has at most 10n51.5849n10^{\frac{n}{5}}\approx 1.5849^n maximal induced matchings, which is attained by the disjoint union of copies of the complete graph K5K_5. In this paper, we show that the maximum number of maximal and maximum induced matchings in a connected graph of order n is \begin{align*} \begin{cases} {n\choose 2} &~ {\rm if}~ 1\leq n\le 8; \\ {{\lfloor \frac{n}{2} \rfloor}\choose 2}\cdot {{\lceil \frac{n}{2} \rceil}\choose 2} -(\lfloor \frac{n}{2} \rfloor-1)\cdot (\lceil \frac{n}{2} \rceil-1)+1 &~ {\rm if}~ 9\leq n\le 13; \\ 10^{\frac{n-1}{5}}+\frac{n+144}{30}\cdot 6^{\frac{n-6}{5}} &~ {\rm if}~ 14\leq n\le 30;\\ 10^{\frac{n-1}{5}}+\frac{n-1}{5}\cdot 6^{\frac{n-6}{5}} & ~ {\rm if}~ n\geq 31, \\ \end{cases} \end{align*} and also show that this bound is tight. This result implies that we can enumerate all maximal induced matchings of an n-vertex connected graph in time O(1.5849n)O(1.5849^n). Moreover, our result provides an estimate on the number of maximal dissociation sets of an n-vertex connected graph.
Chapter
In this paper, we present a faster exact algorithm which solves the Maximum Induced Matching problem for subcubic graphs. Here let n be the overall number of vertices and k be the number of those vertices of degree 3 where all neighbours have also at least degree 2. Then the runtime is at most O(1.2335k)Poly(n)O(1.2335^k) \cdot Poly(n), giving an FPT bound for the time used by the algorithm; the algorithm uses the result of Monien and Preis combined with a bound obtained by applying the measure and conquer technique where the number k replaces n as the measure used; note that knk \le n.
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Chapter
A matching M in a graph G is an acyclic matching if the subgraph of G induced by the endpoints of the edges of M is a forest. Given a graph G and a positive integer \ell , Acyclic Matching asks whether G has an acyclic matching of size (i.e., the number of edges) at least \ell . In this paper, we first prove that assuming W[1]FPT\mathsf {W[1]\nsubseteq FPT}, there does not exist any FPT\textsf{FPT}-approximation algorithm for Acyclic Matching that approximates it within a constant factor when parameterized by \ell . Our reduction is general in the sense that it also asserts FPT\textsf{FPT}-inapproximability for Induced Matching and Uniquely Restricted Matching. We also consider three below-guarantee parameters for Acyclic Matching, viz. n2\frac{n}{2}-\ell , MM(G)\mathsf {MM(G)}-\ell , and IS(G)\mathsf {IS(G)}-\ell , where n is the number of vertices in G, MM(G)\mathsf {MM(G)} is the matching number of G, and IS(G)\mathsf {IS(G)} is the independence number of G. We note that the result concerning the below-guarantee parameter n2\frac{n}{2}-\ell is the most technical part of our paper. Also, we show that Acyclic Matching does not exhibit a polynomial kernel with respect to vertex cover number (or vertex deletion distance to clique) plus the size of the matching unless NPcoNP/poly\textsf{NP}\subseteq \textsf{coNP} /\textsf{poly}.
Chapter
A matching is a subset of edges in a graph G that do not share an endpoint. A matching M is a P\mathcal {P}-matching if the subgraph of G induced by the endpoints of the edges of M satisfies property P\mathcal {P}. For example, if the property P\mathcal {P} is that of being a matching, being acyclic, or being disconnected, then we obtain an induced matching, an acyclic matching, and a disconnected matching, respectively. In this paper, we analyze the problems of the computation of these matchings from the viewpoint of Parameterized Complexity with respect to the parameter treewidth.
Preprint
Structural graph parameters play an important role in parameterized complexity, including in kernelization. Notably, vertex cover, neighborhood diversity, twin-cover, and modular-width have been studied extensively in the last few years. However, there are many fundamental problems whose preprocessing complexity is not fully understood under these parameters. Indeed, the existence of polynomial kernels or polynomial Turing kernels for famous problems such as Clique, Chromatic Number, and Steiner Tree has only been established for a subset of structural parameters. In this work, we use several techniques to obtain a complete preprocessing complexity landscape for over a dozen of fundamental algorithmic problems.
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Given a graph G = (V, E) the Maximum r -Regular Induced Subgraph problem is to find a vertex set R ⊆ V of maximum cardinality such that G[R] is r-regular. An induced matching M ⊆ E in a graph G = (V, E) is a matching such that no two edges in M are joined by any third edge of the graph. The Maximum Induced Matching problem is to find an induced matching of maximum cardinality. By definition the maximum induced matching problem is the maximum 1-regular induced subgraph problem. Gupta et al. gave an o(2n ) time algorithm for solving the Maximum r -Regular Induced Subgraph problem. This algorithm solves the Maximum Induced Matching problem in time O ∗ (1.6957n ) where n is the number of vertices in the input graph. In this paper, we show that the maximum induced matching problem can be reduced to the maximum independent set problem and we give a more efficient algorithm for the Maximum Induced Matching problem running in time O ∗ (1.4786n ).
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Conference Paper
A subset of vertices in a graph is called a dissociation set if it induces a subgraph with a vertex degree of at most 1. The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given graph is known to be NP-hard for bipartite graphs. We show that the maximum dissociation set problem is NP-hard for planar line graphs of planar bipartite graphs. In addition, we describe several polynomially solvable cases for the problem under consideration. One of them deals with the subclass of the so-called chair-free graphs. Furthermore, the related problem of finding a maximal (by inclusion) dissociation set of minimum size in a given graph is studied, and NP-hardness results for this problem, namely for weakly chordal and bipartite graphs, are derived. Finally, we provide inapproximability results for the dissociation set problems mentioned above.
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The Maximum Independent Set problem in d-box graphs, i.e., in the intersection graphs of axis-parallel rectangles in R d , is a challenge open problem. For any fixed d ≥ 2 the problem is NP-hard and no approximation algorithm with ratio o(log d−1 n) is known. In some restricted cases, e.g., for d-boxes with bounded aspect ratio, a PTAS exists [17]. In this paper we prove APX-hardness (and hence non-existence of a PTAS, unless P = NP), of the Maximum Independent Set problem in d-box graphs for any fixed d ≥ 3. We state also first explicit lower bound 443 442 on efficient approximability in such case. Additionally, we provide a generic method how to prove APX-hardness for many NP-hard graph optimization problems in d-box graphs for any fixed d ≥ 3. In 2-dimensional case we give a generic approach to NP-hardness results for these problems in highly restricted intersection graphs of axis-parallel unit squares (alternatively, in unit disk graphs).
Conference Paper
We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. We state the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. For most of dominating set problems we prove asymptotically almost tight lower bounds. The results are applied to improve the lower bounds for other related problems such as the Maximum Induced Matching problem and the Maximum Leaf Spanning Tree problem.
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An induced matching in a graph G is a set of edges, no two of which meet a common node or are joined by an edge of G; that is, an induced matching is a matching which forms an induced subgraph. Induced matchings in graph G correspond precisely to independent sets of nodes in the square of the line-graph of G, which we denote by [L(G)]2. Often, if G has a nice representation as an intersection graph, we can obtain a nice representation of [L(G)]2 as an intersection graph. Then, if the independent set problem is polytime-solvable in [L(G)]2, the induced matching problem is polytime-solvable in G.
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In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio d−1 for MIM in d-regular graphs, for each d⩾3. We also prove that MIM is APX-complete in d-regular graphs, for each d⩾3.
Conference Paper
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We address the problem of efficient data gathering in a wireless network through multihop communication. We focus on two objectives related to flow times, that is, the times spent by data packets in the system: minimization of the maximum flow time and minimization of the average flow time of the packets. For both problems we prove that, unless P&equals;NP, no polynomial-time algorithm can approximate the optimal solution within a factor less than Ω(m1−&epsis;) for any 0m is the number of packets. We then assess the performance of two natural algorithms by proving that their cost remains within the optimal cost of the respective problem if we allow the algorithms to transmit data at a speed 5 times higher than that of the optimal solutions to which we compare them.
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In this paper we investigate the complexity of finding maximum right angle free subsets of a given set of points in the plane. For a set of rational pointsP in the plane, theright angle number ρ(P) (respectivelyrectilinear right angle number ρ R (P)) ofP is the cardinality of a maximum subset ofP, no three members of which form a right angle triangle (respectively a right angle triangle with its side or base parallel to thex-axis). It is shown that both parameters areNP-hard to compute. The latter problem is also shown to be equivalent to finding a minimum dominating set in a bipartite graph. This is used to show that there is a polynomial algorithm for computingρ R (P) whenP is a horizontally-convex subset of the lattice ℤ × ℤ (P ishorizontally-convex if for any pair of points inP which lie on a horizontal line, every lattice point between them is also inP). We then show that this algorithm yields a 1/2-approximate algorithm for the right angle number of a convex subregion of the lattice.
Chapter
Camion proved that every real-valued matrix A can be transformed by pivoting operations and nonzero multiplications of columns into a nonnegative matrix. In this paper we describe a finite algorithm to make this transformation, based on the results of Camion. Our main result is that when A is a totally unimodular matrix this transformation can be made by a polynomial algorithm. Key wordsLinear Algebra–Matroid Theory–Total Unimodularity
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We state that various domination problems are polynomial time solvable in convex bipartite graphs, and we give the main ideas of the algorithms. TOTAL DOMINATING SET is polynomial even for chordal bipartite graphs. Further, it is shown by a reduction from 3SAT that INDEPENDENT DOMINATING SET remains NP-complete when restricted to chordal bipartite graphs.
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A k-cluster in a graph is an induced subgraph on k vertices which maximizes the number of edges. Both the k-cluster problem and the k-dominating set problem are NP-complete for graphs in general. In this paper we investigate the complexity status of these problems on various sub-classes of perfect graphs. In particular, we examine comparability graphs, chordal graphs, bipartite graphs, split graphs, cographs and κ-trees. For example, it is shown that the k-cluster problem is NP-complete for both bipartite and chordal graphs and the independent k-dominating set problem is NP-complete for bipartite graphs. Furthermore, where the k-cluster problem is polynomial we study the weighted and connected versions as well. Similarly we also look at the minimum k-dominating set problem on families which have polynomial k-dominating set algorithms.
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We introduce a new low-degree--test, one that uses the restriction of low-degree polynomials to planes (i.e., affine sub-spaces of dimension 2), rather than the restriction to lines (i.e., affine sub-spaces of dimension 1). We prove the new test to be of a very small errorprobability (in particular, much smaller than constant). The new test enables us to prove a low-error characterization of NP in terms of PCP. Specifically, our theorem states that, for any given ffl ? 0, membership in any NP language can be verified with O(1) accesses, each reading logarithmic number of bits, and such that the error-probability is 2 Gamma log 1Gammaffl n . Our results are in fact stronger, as stated below. One application of the new characterization of NP is that approximating SET-COVER to within a logarithmic factors is NP-hard. Previous analysis for low-degree--tests, as well as previous characterizations of NP in terms of PCP, have managed to achieve, with constant number of accesses, error...