Linear separation of connected dominating sets in graphs

Abstract

A connected dominating set in a graph is a dominating set of vertices that induces a connected subgraph. Following analogous studies in the literature related to independent sets, dominating sets, and total dominating sets, we study in this paper the class of graphs in which the connected dominating sets can be separated from the other vertex subsets by a linear weight function. More precisely, we say that a graph is connected-domishold if it admits non-negative real weights associated to its vertices such that a set of vertices is a connected dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We characterize the graphs in this non-hereditary class in terms of a property of the set of minimal cutsets of the graph. We give several characterizations for the hereditary case, that is, when each connected induced subgraph is required to be connected-domishold. The characterization by forbidden induced subgraphs implies that the class properly generalizes two well known classes of chordal graphs, the block graphs and the trivially perfect graphs. Finally, we study certain algorithmic aspects of connected-domishold graphs. Building on connections with minimal cutsets and properties of the derived hypergraphs and Boolean functions, we show that our approach leads to new polynomially solvable cases of the weighted connected dominating set problem.

Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1‐Sperner hypergraphs

“…We conclude this section by stating two related theorems due to Chiarelli and Milanič [12,13] about the structure of graphs defined similarly as domishold graphs, but with respect to total, resp. connected domination.…”

“…Second, a common approach of applying hypergraph theory to graphs is to define and study a hypergraph derived in an appropriate way from a given graph, depending on what type of property of a graph or its vertex or edge subsets one is interested in. This includes matching hypergraphs [1,63], various clique [32,41,47,51,63], independent set [32,63], neighborhood [12,27], separator [13,60], and dominating set hypergraphs [12,13], etc. Close interrelations between hypergraphs and monotone Boolean functions can be useful in such studies, 1 arXiv:1805.03405v3 [math.CO] 29 May 2018 allowing for the transfer and applications of results from the theory of Boolean functions (see, e.g., [21]).…”

“…dominating sets from the characteristic vectors of all other sets. The corresponding hypergraphs derived from a given graph, obtained using the notion of transversal hypergraphs, are the hypergraphs of inclusion-minimal vertex covers and inclusion-minimal closed neighborhoods, respectively.More recent examples involve the works of Chiarelli and Milanič [11,12] who made use of a slightly more general class than 1-Sperner hypergraphs, called dually Sperner hypergraphs, to characterize two classes of graphs defined by the following properties: every induced subgraph has a non-negative linear vertex weight function separating the characteristic vectors of all total dominating sets [11], resp. connected dominating sets [12], from the characteristic vectors of all other sets.…”

“…In these cases, the corresponding transversal hypergraphs derived from a graph G are the hypergraphs of inclusion-minimal (open) neighborhoods and inclusion-minimal cutsets, respectively. Due to the close relation between 1-Sperner and dually Sperner hypergraphs (see [9]), all the results from [11,12] can be equivalently stated using 1-Sperner hypergraphs. In particular, the results of the extended abstract [11] are stated using the 1-Sperner property in the full version of the paper [13].A common approach in these studies is to consider various hypergraphs associated to a given graph and investigate the class of graphs for which the associated hypergraph is threshold.…”

“…Due to the close relation between 1-Sperner and dually Sperner hypergraphs (see [9]), all the results from [11,12] can be equivalently stated using 1-Sperner hypergraphs. In particular, the results of the extended abstract [11] are stated using the 1-Sperner property in the full version of the paper [13].A common approach in these studies is to consider various hypergraphs associated to a given graph and investigate the class of graphs for which the associated hypergraph is threshold. The goal is then to characterize these graph classes and, in particular, to understand under what conditions they are hereditary (that is, it is closed under vertex deletion).…”