Partitioning the Vertices of a Graph into Two Total Dominating Sets

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every selfcomplementary graph with minimum degree at least two has such a partition.     

On the Convexity Number of Graphs

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G. We prove NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G) ≥ k. Furthermore, we identify natural necessary extension properties of graphs of small convexity number and study the interplay between these properties and upper bounds on the convexity number.     

Domination versus disjunctive domination in graphs

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number of G is the minimum cardinality of a dominating set of G. For a positive integer b, a set S of vertices in a graph G is a b-disjunctive dominating set in G if every vertex v not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it in G. The b-disjunctive domination number of G is the minimum cardinality of a b-disjunctive dominating set. In this paper, we continue the study of disjunctive domination in graphs. We present properties of b-disjunctive dominating sets in a graph. A characterization of minimal b-disjunctive dominating sets is given. We obtain bounds on the ratio of the domination number and the b-disjunctive domination number for various families of graphs, including regular graphs and trees.     

Cyclability of 3‐connected graphs

A subset S of vertices of a graph G is called cyclable in G if there is in G some cycle containing all the vertices of S. We denote by α(S, G) the number of vertices of a maximum independent set of G[S]. We prove that if G is a 3‐connected graph or order n and if S is a subset of vertices such that the degree sum of any four independent vertices of S is at least n + 2α(S, G) −2, then S is cyclable. This result implies several known results on cyclability or Hamiltonicity. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 191–203, 2000     

Hamiltonicity and reversing arcs in digraphs

In this paper we introduce a new hamiltonian-like property of graphs. A graph G is said to be cyclable if for each orientation D of G there is a set S of vertices such that reversing all the arcs of D with one end in S results in a hamiltonian digraph. We characterize cyclable complete multipartite graphs and prove that the fourth power of any connected graph G with at least five vertices is cyclable. If, moreover, G is two-connected then its cube is cyclable. These results are shown to be best possible in a sense. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 13–30, 1998     

Two characterizations of chain partitioned probe graphs

Chain graphs are exactly bipartite graphs without induced 2K ₂ (a graph with four vertices and two disjoint edges). A graph G=(V,E) with a given independent set S⊆V (a set of pairwise non-adjacent vertices) is said to be a chain partitioned probe graph if G can be extended to a chain graph by adding edges between certain vertices in S. In this note we give two characterizations for chain partitioned probe graphs. The first one describes chain partitioned probe graphs by six forbidden subgraphs. The second one characterizes these graphs via a certain “enhanced graph”: G is a chain partitioned probe graph if and only if the enhanced graph G ^* is a chain graph. This is analogous to a result on interval (respectively, chordal, threshold, trivially perfect) partitioned probe graphs, and gives an O(m ²)-time recognition algorithm for chain partitioned probe graphs.     

Packing in regular graphs

A set S of vertices in a graph G is a packing if the vertices in S are pairwise at distance at least 3 apart in G. The packing number of G, denoted by ρ(G), is the maximum cardinality of a packing in G. Favaron [Discrete Math. 158 (1996), 287–293] showed that if G is a connected cubic graph of order n different from the Petersen graph, then ρ(G) ≥ n/8. In this paper, we generalize Favaron’s result. We show that for k ≥ 3, if G is a connected k-regular graph of order n that is not a diameter-2 Moore graph, then ρ(G) ≥ n/(k² ? 1).     

The orders of graphs with prescribed degree sets

The degree set ??^G of a graph G is the set of degrees of the vertices of G. For a finite nonempty set S of positive integers, all positive integers p are determined for which there exists a graph G of order p such that ??^G = S.     

Inequality of Nordhaus-Gaddum type for total outer-connected domination in graphs

A set S of vertices in a graph G = (V, E) without isolated vertices is a total outer-connected dominating set (TCDS) of G if S is a total dominating set of G and G[V − S] is connected. The total outer-connected domination number of G, denoted by γ _tc (G), is the minimum cardinality of a TCDS of G. For an arbitrary graph without isolated vertices, we obtain the upper and lower bounds on γ _tc (G) + γ _tc ($ \bar G $ \bar G ), and characterize the extremal graphs achieving these bounds.     

Steiner Numbers in Graphs

Abstract

The Steiner distance d(S) of a set S of vertices in a connected graph G is the minimum size of a connected subgraph of G that contains S. The Steiner number s(G) of a connected graph G of order p is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = p—1. A smallest set S of vertices of a connected graph G of order p for which d(S) = p—1 is called a Steiner spanning set of G. It is shown that every connected graph has a unique Steiner spanning set. If G is a connected graph of order p and k is an integer with 0 ≤ k ≤ p—1, then the kth Steiner number sk(G) of G is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = k. The sequence s_o(G),s₁ (G),…,8_p-1(G) is called the Steiner sequence of G. Steiner sequences for trees are characterized.     

Total Domination in Graphs with Given Girth

A set S of vertices in a graph G without isolated vertices is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ _t (G) of G. In this paper, we establish an upper bound on the total domination number of a graph with minimum degree at least two in terms of its order and girth. We prove that if G is a graph of order n with minimum degree at least two and girth g, then γ _t (G) ≤ n/2 + n/g, and this bound is sharp. Our proof is an interplay between graph theory and transversals in hypergraphs. Michael A. Henning: Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Connected Domination Stable Graphs Upon Edge Addition

Abstract

A set S of vertices in a graph G is a connected dominating set of G if S dominates G and the subgraph induced by S is connected. We study the graphs for which adding any edge does not change the connected domination number.     

On the Independent Domination Number of Regular Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs.     

Edge-connectivity and edge-superconnectivity in sequence graphs

Given an integer k?1 and any graph G, the sequence graph S_k(G) is the graph whose set of vertices is the set of all walks of length k in G. Moreover, two vertices of S_k(G) are joined by an edge if and only if their corresponding walks are adjacent in G.In this paper we prove sufficient conditions for a sequence graph S_k(G) to be maximally edge-connected and edge-superconnected depending on the parity of k and on the vertex-connectivity of the original graph G.     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

A survey of selected recent results on total domination in graphs

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. In this paper, we offer a survey of selected recent results on total domination in graphs.     

The disjunctive domination number of a graph

Abstract

For a positive integer b, we define a set S of vertices in a graph G as a b-disjunctive dominating set if every vertex not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it. The b-disjunctive domination number is the minimum cardinality of such a set. This concept is motivated by the concepts of distance domination and exponential domination. In this paper, we start with some simple results, then establish bounds on the parameter especially for regular graphs and claw-free graphs. We also show that determining the parameter is NP-complete, and provide a linear-time algorithm for trees.     

A note on the computational complexity of graph vertex partition

A stable set of a graph is a vertex set in which any two vertices are not adjacent. It was proven in [A. Brandstädt, V.B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability, Discrete Appl. Math. 89 (1998) 59-73] that the following problem is NP-complete: Given a bipartite graph G, check whether G has a stable set S such thatG-Sis a tree. In this paper we prove the following problem is polynomially solvable: Given a graph G with maximum degree 3 and containing no vertices of degree 2, check whether G has a stable set S such thatG-Sis a tree. Thus we partly answer a question posed by the authors in the above paper. Moreover, we give some structural characterizations for a graph G with maximum degree 3 that has a stable set S such that G-S is a tree.     

Locating and paired-dominating sets in graphs

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater [T.W. Haynes, P.J. Slater, Paired-domination in graphs, Networks 32 (1998), 199–206]. A paired-dominating set of a graph G with no isolated vertex is a dominating set S of vertices whose induced subgraph has a perfect matching. We consider paired-dominating sets which are also locating sets, that is distinct vertices of G are dominated by distinct subsets of the paired-dominating set. We consider three variations of sets which are paired-dominating and locating sets and investigate their properties.     

Steiner centers in graphs

The Steiner distance of a set S of vertices in a connected graph G is the minimum size among all connected subgraphs of G containing S. For n ≥ 2, the n-eccentricity e_n(ν) of a vertex ν of a graph G is the maximum Steiner distance among all sets S of n vertices of G that contains ν. The n-diameter of G is the maximum n-eccentricity among the vertices of G while the n-radius of G is the minimum n-eccentricity. The n-center of G is the subgraph induced by those vertices of G having minimum n-eccentricity. It is shown that every graph is the n-center of some graph. Several results on the n-center of a tree are established. In particular, it is shown that the n-center of a tree is a tree and those trees that are n-centers of trees are characterized.