Sandra M. Hedetniemi

Clemson University, Anderson, Indiana, United States

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Publications (73)26.36 Total impact

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    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.
    Parallel Processing Letters 03/2013; 23(1).
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    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.
    JCMCC. The Journal of Combinatorial Mathematics and Combinatorial Computing. 01/2013; 85.
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    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.
    AKCE International Journal of Graphs and Combinatorics. 01/2013; 10(4).
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    AKCE International Journal of Graphs and Combinatorics. 01/2013; 10(1).
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    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.
    Information Processing Letters 08/2012; 112(16):621–623. · 0.48 Impact Factor
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    ABSTRACT: A set S of vertices in a graph G=(V,E) is a dominating set if every vertex in V⊂S is adjacent to at least one vertex in S. A vertex v in a dominating set S is said to be cost effective if it is adjacent to at least as many vertices in V⊂S as it is in S, and is very cost effective if it is adjacent to more vertices in V⊂S than to vertices in S. A dominating set S is (very) cost effective if every vertex in S is (very) cost effective. In this paper we introduce the study of cost effective domination in graphs and relate it to the previously studied unfriendly partitions of graphs.
    Congressus Numerantium. 01/2012;
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    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.
    JCMCC. The Journal of Combinatorial Mathematics and Combinatorial Computing. 01/2012; 80.
  • Gerd H. Fricke, Sandra Mitchell Hedetniemi, Stephen T. Hedetniemi, Kevin R. Hutson
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    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.
    Discussiones Mathematicae Graph Theory. 01/2011; 31:517-531.
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    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.
    Discrete Applied Mathematics 01/2011; 159:15-22. · 0.68 Impact Factor
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    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 all minimal (or maximal) sets S of the given type, where |V|<|S|≥2, have the property that every vertex e∈V-S is within distance at most x to a vertex u∈S (shortest distance), and within distance at most y to a second vertex w∈S (second shortest 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 (the internal distance). A dominating set S⊆V in a graph G is a set having the property that every vertex v∈V-S is within distance 1 to sone vertex in S, or equivalently, whose shortest distance x=1. In this paper we determine the secondary distances y and internal distances z for 31 types of sets in graphs, whose shortest distance x=1.
    AKCE International Journal of Graphs and Combinatorics. 01/2009; 6(2).
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    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.
    Journal of Combinatorics, Information & System Sciences. 01/2009; 34(1).
  • Wayne Goddard, Sandra M. Hedetniemi, Stephen T. Hedetniemi
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    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.
    JCMCC. The Journal of Combinatorial Mathematics and Combinatorial Computing. 01/2009; 68.
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    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.
    AKCE International Journal of Graphs and Combinatorics. 01/2008; 5(2).
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    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.
    Ars Combinatoria -Waterloo then Winnipeg- 01/2008; 86. · 0.20 Impact Factor
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    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.
    Bulletin of the Institute of Combinatorics and its Applications. 01/2008; 54.
  • Sandra M. Hedetniemi, Stephen T. Hedetniemi, Arthur L. Liestman
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    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.
    Networks 10/2006; 18(4):319 - 349. · 0.74 Impact Factor
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    ABSTRACT: Let G(V,E) be a graph, X⊆V. Set B(X)={v∈V∖X∣v has a neighbour in X} and ∂(X)=|B(X)|-|X|. Call max{∂(X)∣X⊆V} the differential of G. This is used to model a token game on a graph described in the abstract. The authors also introduce the I-, A-, and B-differentials by replacing |X| in the definition of ∂ by |I(X)|=|X∖N(X)|, |A(X)|=|X∩N(X)| and 0. All these concepts have been continued by different authors under different names. The authors give bounds for these differentials for paths, circuits, and trees. Moreover, there are constructed trees with several prescribed values of ∂.Reviewer: 00000492 Ulrich Knauer (Oldenburg)
    Utilitas Mathematica 01/2006; 69. · 0.32 Impact Factor
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    ABSTRACT: We say that a function f:V→{0,1,…,diam(G)}f:V→{0,1,…,diam(G)} is a broadcast if for every vertex v∈Vv∈V, f(v)⩽e(v)f(v)⩽e(v), where diam(G)diam(G) denotes the diameter of G and e(v)e(v) denotes the eccentricity of vv. The cost of a broadcast is the value f(V)=∑v∈Vf(v). In this paper we introduce and study the minimum and maximum costs of several types of broadcasts in graphs, including dominating, independent and efficient broadcasts.
    Discrete Applied Mathematics 01/2006; 154:59-75. · 0.68 Impact Factor
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    Wayne Goddard, Sandra Mitchell Hedetniemi, Stephen T. Hedetniemi, Renu Laskar
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    ABSTRACT: Abstract For a graph property P, we dene a P-matching as a set M of disjoint edges such that the subgraph induced by the vertices incident to M has property P. Previous examples include strong/induced matchings and uniquely restricted matchings. We explore the general properties of P-matchings, but especially the cases where P is the property of being acyclic or the property of being disconnected. We consider bounds on and the complexity of the maximum cardinality of a P-matching and the minimum cardinality of a maximal P-matching. Key words: matchings, graphs, induced matchings
    Discrete Mathematics 04/2005; 293:129-138. · 0.57 Impact Factor
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    ABSTRACT: A de-pair (ode-pair) in a graph consists of two disjoint subsets of vertices with the same closed neighborhood (open neighborhood). We consider the question of determining the smallest and largest subsets over all such pairs. We provide sharp bounds on these for general graphs and for trees, and show that the associated parameters are computable for trees but intractable in general.
    AKCE International Journal of Graphs and Combinatorics. 01/2005; 2(2).