[Show abstract][Hide abstract]ABSTRACT: In the study of domination in graphs, relationships between the concepts of maximal independent sets, minimal dominating sets and maximal irredundant sets are used to establish what is known as the domination chain of parameters: ir(G) <= gamma(G) <= i(G) <= beta(0) (G) <= Gamma(G) <= IR(G). This chain of inequalities has become one of the major focal points in domination theory. Starting from the concept of vertex cover, we introduce six graph-theoretic parameters which obey a chain of inequalities, which we call the covering chain of the graph G. We prove that this covering chain is the dual of the domination chain.
[Show abstract][Hide abstract]ABSTRACT: For a graph \(G=(V,E)\) , a Roman dominating function \(f:V\rightarrow \{0,1,2\}\) has the property that every vertex \(v\in V\) with \(f(v)=0\) has a neighbor \(u\) with \(f(u)=2\) . The weight of a Roman dominating function \(f\) is the sum \(f(V)=\sum \nolimits _{v\in V}f(v)\) , and the minimum weight of a Roman dominating function on \(G\) is the Roman domination number of \(G\) . In this paper, we define the Roman independence number, the upper Roman domination number and the upper and lower Roman irredundance numbers, and then develop a Roman domination chain parallel to the well-known domination chain. We also develop sharpness, strictness and bounds for the Roman domination chain inequalities.
Full-text Article · Mar 2015 · Graphs and Combinatorics
[Show abstract][Hide abstract]ABSTRACT: An unfriendly partition is a partition of the vertices of a graph G=(V,E) into two sets, say Red R(V) and Blue B(V), such that every Red vertex has at least as many Blue neighbors as Red neighbors, and every Blue vertex has at least as many Red neighbors as Blue neighbors. We present three polynomial time, self-stabilizing algorithms for finding unfriendly partitions in arbitrary graphs G, or equivalently into two disjoint dominating sets.
[Show abstract][Hide abstract]ABSTRACT: Let G = (V, E) be a graph. A partition π = {V1, V2,., Vk} of the vertex set V of G into k color classes Vi, with 1 ≤ i ≤ k, is called a quorum coloring if for every vertex v ∈ V, at least half of the vertices in the closed neighborhood N[v] of v have the same color as v. In this paper we introduce the study of quorum colorings of graphs and show that they are closely related to the concept of defensive alliances in graphs. Moreover, we determine the maximum quorum coloring of a hypercube.
Full-text Article · Jan 2013 · AKCE International Journal of Graphs and Combinatorics
[Show abstract][Hide abstract]ABSTRACT: Let G=(V,E) be a graph and let N(v)={u:uv∈E} be the open neighborhood of a vertex v∈V . The degree of v, deg (v)=|N(v)|, equals the number of vertices u that are adjacent to v. By considering the relationships between deg (v) and the degrees deg (u), for every u∈N(v), we define 10 types of vertices and study some of their properties.
Full-text Article · Jan 2013 · AKCE International Journal of Graphs and Combinatorics
[Show abstract][Hide abstract]ABSTRACT: Given a set S⊆V in a graph G=(V,E), we say that a vertex v∈V is perfect if |N[v]∩S|=1, that is, the closed neighborhood N[v]={v}∪{u∣uv∈E} of v contains exactly one vertex in S. A vertex v is almost perfect if it is either perfect or is adjacent to a perfect vertex. Similarly, we can say that a set S⊂V is (almost) perfect if every vertex v∈S is (almost) perfect; S is externally (almost) perfect if every vertex u∈V-S is (almost) perfect; and S is completely (almost) perfect if every vertex v∈V is (almost) perfect. In this paper we relate these concepts of perfection to independent sets, dominating sets, efficient and perfect dominating sets, distance-2 dominating sets, and to perfect neighborhood sets in graphs. The concept of a set being almost perfect also provides an equivalent definition of irredundance in graphs.
Article · Jan 2013 · Journal of Combinatorial Mathematics and Combinatorial Computing
[Show abstract][Hide abstract]ABSTRACT: The efficiency of a set S⊆VS⊆V in a graph G=(V,E)G=(V,E), is defined as ε(S)=|{v∈V−S:|N(v)∩S|=1}|ε(S)=|{v∈V−S:|N(v)∩S|=1}|; in other words, the efficiency of a set S equals the number of vertices in V−SV−S that are adjacent to exactly one vertex in S. A set S is called optimally efficient if for every vertex v∈V−Sv∈V−S, ε(S∪{v})⩽ε(S)ε(S∪{v})⩽ε(S), and for every vertex u∈Su∈S, ε(S−{u})<ε(S)ε(S−{u})<ε(S). We present a polynomial time self-stabilizing algorithm for finding an optimally efficient set in an arbitrary graph. This algorithm is designed using the distance-2 self-stabilizing model of computation.
Article · Aug 2012 · Information Processing Letters
[Show abstract][Hide abstract]ABSTRACT: Let G=(V,E) be an undirected graph and let π={V 1 ,V 2 ,⋯,V k } be a partition of the vertices V of G into k blocks V i . From this partition one can construct the following digraph D(π)=(π,E(π)), the vertices of which correspond one-to-one with the k blocks V i of π, and there is an arc from V i to V j if every vertex in V j is adjacent to at least one vertex in V i , that is, V i dominates V j . We call the digraph D(π) the domination digraph of π. A triad is one of the 16 digraphs on three vertices having no loops or multiple arcs. In this paper we study the algorithmic complexity of deciding if an arbitrary graph G has a given digraph as one of its domination digraphs, and in particular, deciding if a given triad is one of its domination digraphs. This generalizes results for the domatic number.
Article · Jan 2012 · Journal of Combinatorial Mathematics and Combinatorial Computing
[Show abstract][Hide abstract]ABSTRACT: We present a collection of new structural, algorithmic, and complexity results for matching problems of two types. The first problem involves the computation of k-maximal matchings, where a matching is k-maximal if it admits no augmenting path with ≤2k vertices. The second involves finding a maximal set of vertices that is matchable — comprising one side of the edges in some matching. Among our results, we prove that the minimum cardinality β2 of a 2-maximal matching is at most the minimum cardinality μ of a maximal matchable set, with equality attained for triangle-free graphs. We show that the parameters β2 and μ are NP-hard to compute in bipartite and chordal graphs, but can be computed in linear time on a tree. Finally, we also give a simple linear-time algorithm for finding a 3-maximal matching, a consequence of which is a simple linear-time 3/4-approximation algorithm for the maximum-cardinality matching problem in a general graph.
Full-text Article · Jan 2011 · Discrete Applied Mathematics
[Show abstract][Hide abstract]ABSTRACT: A set S⊆V is a dominating set of a graph G=(V,E) if every vertex in V-S is adjacent to at least one vertex in S. The domination number γ(G) of G equals the minimum cardinality of a dominating set S in G; we say that such a set S is a γ-set. In this paper we consider the family of all γ-sets in a graph G and we define the γ-graph G(γ)=(V(γ),E(γ)) of G to be the graph whose vertices V(γ) correspond 1-to-1 with the γ-sets of G, and two γ-sets, say D 1 and D 2 , are adjacent in E(γ) if there exists a vertex v∈D 1 and a vertex w∈D 2 such that v is adjacent to w and D 1 =D 2 -{w}∪{v}, or equivalently, D 2 =D 1 -{v}∪{w}. In this paper we initiate the study of γ-graphs of graphs.
Article · Jan 2011 · Discussiones Mathematicae Graph Theory
[Show abstract][Hide abstract]ABSTRACT: We introduce a k-response set as a set of vertices where responders can be placed so that given any set of k emergencies, these responders can respond, one per emergency, where each responder covers its own vertex and its neighbors. A weak k-response set does not have to worry about emergencies at the vertices of the set. We define Rk, and rk as the minimum cardinality of such sets. We provide bounds on these parameters and discuss connections with domination invariants. For example, for a graph G of order n and minimum degree at least 2, R2(G) ≤ 2n/3, while r2(G) ≤ n/2 provided G is also connected and not K3. We also provide bounds for trees T of order n. We observe that there are for each k trees for which rk(T) ≤ n/2, but that the minimum Rk(T) appears to grows with k; a novel computer algorithm is used to show that R3(T) > n/2. As expected, these parameters are NP-hard to compute, and we provide a linear-time algorithm for trees for fixed k.
Full-text Article · Feb 2009 · Journal of Combinatorial Mathematics and Combinatorial Computing
[Show abstract][Hide abstract]ABSTRACT: For any given type of a set of vertices in a connected graph G=(V,E), we seek to determine the smallest integers (x,y:z) such that for all minimal (or maximal) sets S of the given type, where |V|>|S|≥2, every vertex v∈V-S is within shortest distance at most x to a vertex u∈S (called dominating distance), and within distance at most y to a second vertex w∈S (called secondary distance). We also seek to determine the smallest integer z such that every vertex u∈S is within distance at most z to a closest neighbor w∈S (called internal distance). In this paper, a sequel to two previous papers [S. M. Hedetniemi et al., ibid. 5, No. 2, 103–115 (2008; Zbl 1176.05055); ibid. 6, No. 2, 239–266 (2009; Zbl 1210.05032)], we determine the secondary and internal distances (2,y:z) for 16 types of sets, all of which are distance-2 dominating sets, that is, whose dominating distances are at most 2.
Article · Jan 2009 · AKCE International Journal of Graphs and Combinatorics
[Show abstract][Hide abstract]ABSTRACT: We introduce a k-response set as a set of vertices where responders can be placed so that given any set of k emergencies, these responders can respond, one per emergency, where each responder covers its own vertex and its neighbors. A weak k-response set does not have to worry about emergencies at the vertices of the set. We define Rk and rk as the minimum cardinality of such sets. We provide bounds on these parameters and discuss connections with domination invariants. For example, for a graph G of order n and minimum degree at least 2, R2(G) � 2n/3, while r2(G) � n/2 provided G is also connected and not K3. We also provide bounds for trees T of order n. We observe that there are for each k trees for which rk(T) � n/2, but that the minimum Rk(T) appears to grows with k; a novel computer algorithm is used to show that R3(T) > n/2. As expected, these parameters are NP-hard to compute, and we provide a linear-time algorithm for trees for fixed k.
Article · Jan 2009 · Journal of Combinatorial Mathematics and Combinatorial Computing
[Show abstract][Hide abstract]ABSTRACT: A vertex v∈Vin a graph G=(V,E) dominates a set S⊆V if it is adjacent to every vertex w∈S, in which case we say that v is a dominator of S. A partition π={V 1 ,V 2 ,⋯,V k } of V(G) is called a dominator partition if every vertex v∈V is a dominator of at least one block V j of π. The dominator partition number of a graph G, denoted π d (G), is the minimum order of a dominator partition of G. In this paper we introduce the concept of dominator partitions and obtain tight bounds for π d (G) for any graph G.
[Show abstract][Hide abstract]ABSTRACT: A function � : V ! {1,...,k} is a broadcast coloring of order k if �(u) = �(v) implies that the distance between u and v is more than �(u). The minimum order of a broadcast coloring is called the broadcast chromatic number of G, and is denotedb(G). In this pa- per we introduce this coloring and study its properties. In particular, we explore the relationship with the vertex cover and chromatic num- bers. While there is a polynomial-time algorithm to determine whether �b(G) � 3, we show that it is NP-hard to determine ifb(G) � 4. We also determine the maximum broadcast chromatic number of a tree, and show that the broadcast chromatic number of the infinite grid is finite.
Full-text Article · Jan 2008 · Ars Combinatoria -Waterloo then Winnipeg-
[Show abstract][Hide abstract]ABSTRACT: Given a dominating set S⊆V in a graph G=(V,E), place one guard at each vertex in S. Should there be a problem at a vertex v∈V-SA, we can send a guard at a vertex it u∈S adjacent to v to handle the problem. If for some reason this guard needs assistance, a second guard can be sent from S to v, but the question is: how long will it take for a second guard to arrive? This is the issue of what we call secondary domination. We focus primarily on dominating sets in which a second guard can arrive in at most two time steps. A (1,2)-dominating set in a graph G=(V,E) is a set S having the property that for every vertex v∈V-S there is at least one vertex in S at distance 1 from v and a second vertex in S at distance at most 2 from v. We present a variety of results about secondary domination, relating this to several other well-studied types of domination.
Full-text Article · Jan 2008 · AKCE International Journal of Graphs and Combinatorics
[Show abstract][Hide abstract]ABSTRACT: An irredundant coloring of a graph G=(V,E) is a partition of V(G) into irredundant sets. The irratic number of a graph G equals the minimum order of an irredundant coloring of G. In this paper we introduce the study of irredundant colorings and the irratic number of a graph.
[Show abstract][Hide abstract]ABSTRACT: Gossiping and broadcasting are two problems of information dissemination described for a group of individuals connected by a communication network. In gossiping every person in the network knows a unique item of information and needs to communicate it to everyone else. In broadcasting one individual has an item of information which needs to be communicated to everyone else. We review the results that have been obtained on these and related problems.
[Show abstract][Hide abstract]ABSTRACT: Let G = (V, E) be an arbitrary graph, and consider the following game. You are allowed to buy as many tokens as you like, say k tokens, at a cost of $1 each. You then place the tokens on some subset of k vertices of V. For each vertex of G which has no token on it, but is adjacent to a vertex with a token on it, you receive $1. Your objective is to maximize your profit, that is, the total value received minus the cost of the tokens bought. Let B(X) be the set of vertices in V - X that have a neighbor in a set X. Based on this game, we define the differential of a set X to be phi(X) = vertical bar B(X)vertical bar - vertical bar X vertical bar, and the differential of a graph to equal the max{phi(X)} for any subset X of V. In this paper, we introduce several different variations of the differential of a graph and study bounds on, and properties of, these novel parameters.