Linear separation of connected dominating sets in graphs

TL;DR

This paper introduces and characterizes the class of connected-domishold graphs, where connected dominating sets can be linearly separated from other vertex subsets, and explores their properties and algorithmic implications.

Contribution

It provides a characterization of connected-domishold graphs via minimal cutsets and forbidden subgraphs, extending known classes like block graphs and trivially perfect graphs.

Findings

Characterization of connected-domishold graphs using minimal cutsets.

Identification of forbidden induced subgraphs for hereditary cases.

Development of polynomial algorithms for weighted connected dominating set problem.

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.