The search of spanning trees with interesting disjunction properties has led to the introduction of edge-disjoint spanning trees, independent spanning trees and more recently completely independent spanning trees. We group together these notions by defining (i, j)-disjoint spanning trees, where i (j, respectively) is the number of vertices (edges, respectively) that are shared by more than one tree. We illustrate how (i, j)-disjoint spanning trees provide some nuances between the existence of disjoint connected dominating sets and completely independent spanning trees. We prove that determining if there exist two (i, j)-disjoint spanning trees in a graph G is NP-complete, for every two positive integers i and j. Moreover we prove that for square of graphs, k-connected interval graphs, complete graphs and several grids, there exist (i, j)-disjoint spanning trees for interesting values of i and j.
Some subsets of vertices @math of a graph @math are @math if @math are pairwise disjoint and each subset is a connected dominating set in @math . There are some works about disjoint connected dominating sets that can be transcribed in terms of internally vertex-disjoint spanning trees (the disjoint connected dominating sets can be used to provide the inner vertices of internally vertex-disjoint spanning trees). The maximum number of disjoint connected dominating sets in a graph @math is the . This parameter is denoted by @math and has been introduced by Hedetniemi and Laskar @cite_0 in 1984. An interesting result about connected domatic number concerns planar graphs, for which Hartnell and Rall have proven that, except @math (which has connected domatic number @math ), their connected domatic number is bounded by 3 @cite_6 . The problem of constructing a connected dominating set is often motivated by wireless ad-hoc networks @cite_7 @cite_3 for which connected dominating sets are used to create a virtual backbone in the network.
@cite_0 @cite_3 @cite_7 @cite_6
1515157528 1495272862 1552743905 2052665499
A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices, @math the minimum degree, and @math the maximum degree. We show that every graph has a domatic partition with @math dominating sets and, moreover, that such a domatic partition can be found in polynomial-time. This implies a @math -approximation algorithm for domatic number, since the domatic number is always at most @math . We also show this to be essentially best possible. Namely, extending the approximation hardness of set cover by combining multiprover protocols with zero-knowledge techniques, we show that for every @math , a @math -approximation implies that @math . This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better. We also show that every graph has a domatic partition with @math dominating sets, where the "o(1)" term goes to zero as @math increases. This can be turned into an efficient algorithm that produces a domatic partition of @math sets. A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3. All graphs considered in this paper are finite graphs without loops and multiple edges. The domatic number of a graph was defined by E. J. Cockayne and S. T. Hedetniemi [1]. Later some related concepts were introduced. The same authors together with R. M. Dawes [2] have introduced the total domatic number; R. Laskar and S. T. Hedetniemi [3] have introduced the connected domatic number. A dominating set (or a total dominating set) in an undirected graph G is a subset D of the vertex set V(G) of G with the property that to each vertex x e V(G) — D (or to each vertex xeV(G) respectively) there exists a vertex y e D adjacent to x. A connected dominating set of G is a dominating set of G with the property that the subgraph of G induced by it is connected. A domatic (or total domatic, or connected domatic) partition of G is a partition of V(G), all of whose classes are dominating (or total dominating, or connected dominating, respectively) sets of G. The maximum number of classes of a domatic (or total domatic, or connected domatic) partition of G is called the domatic (or total domatic, or connected domatic, respectively) number of G. The domatic number of G is denoted by d(G), its total domatic number by <4(G), its connected domatic number by dc(G). The connected domatic number of a graph is well defined only for connected graphs; in a disconnected graph there exists no connected dominating set and thus no connected domatic partition, while in every connected graph there exists at least one connected domatic partition, namely that which consists of one class. The connected domatic number of G is closely related to the vertex connectivity number of G. If G is a connected graph, then a vertex cut of G is a subset R of V(G) with the property that the subgraph of G induced by V(G) — R is disconnected. If G is not a complete graph, then the vertex connectivity number x(G) is the minimum cardinality of a vertex cut of G. If G is a complete graph (i. e. without vertex cuts) with n vertices, then we put x(G) = n — 1. Lemma. Let G be a connected graph which is not complete, let R be its vertex cut, let D be its connected dominating set. Then DnR^0. A set L@?V(G) of a graph G=(V,E) is a liar's dominating set if (1) for all v@?V(G), |N"G[v]@?L|>=2 and (2) for every pair u,v@?V(G) of distinct vertices, |(N"G[u]@?N"G[v])@?L|>=3. A graph G=(V,E) admits a liar's dominating set if each of its connected component has at least three vertices. Given a graph G=(V,E) and an integer K, the liar's domination decision problem (LR-DOMDP) is to decide whether G has a liar's dominating set of cardinality at most K. Slater [P.J. Slater, Liar's Domination, Networks, 54(2) (2009) 70-74] proved that the LR-DOMDP is NP-complete for general graphs. Subsequently, Roden and Slater [M.L. Roden and P.J. Slater, Liar's domination in graphs, Discrete Math., 309(19) (2009) 5884-5890] showed a more general family of problems to each be NP-complete for bipartite graphs. Besides this result, no other algorithmic result for the liar's dominating set problem is available in the literature. In this paper, we first strengthen the complexity result of the LR-DOMDP by showing that this problem remains NP-complete for split graphs and hence for chordal graphs. Finally, we propose a linear time algorithm for computing a minimum cardinality liar's dominating set in a tree.