A note on connected dominating sets of distance-hereditary graphs

A vertex subset of a graph is a dominating set if every vertex of the graph belongs to the set or has a neighbor in it. A connected dominating set is a dominating set such that the induced subgraph of the set is a connected graph. A graph is called distance-hereditary if every induced path is a shortest path. In this note, we give a complete description of the (inclusionwise) minimal connected dominating sets of connected distance-hereditary graphs in the following sense: If G is a connected distance-hereditary graph that has a dominating vertex, any minimal connected dominating set is a single vertex or a pair of two adjacent vertices. If G does not have a dominating vertex, the subgraphs induced by any two minimal connected dominating sets are isomorphic. In particular, any inclusionwise minimal connected dominating set of a connected distance-hereditary graph without dominating vertex has minimal size. In other words, connected distance-hereditary graphs without dominating vertex are connected well-dominated. Furthermore, we show that if G is a distance-hereditary graph that has a minimal connected dominating set X of size at least 2, then for any connected induced subgraph H it holds that the subgraph induced by any minimal connected dominating set of H is isomorphic to an induced subgraph of G[X]

A note on connected dominating sets of distance-hereditary graphs

A vertex subset of a graph is a dominating set if every vertex of the graph belongs to the set or has a neighbor in it. A connected dominating set is a dominating set such that the induced subgraph of the set is a connected graph. A graph is called distance-hereditary if every induced path is a shortest path. In this note, we give a complete description of the (inclusionwise) minimal connected dominating sets of connected distance-hereditary graphs in the following sense: If G is a connected distance-hereditary graph that has a dominating vertex, any minimal connected dominating set is a single vertex or a pair of two adjacent vertices. If G does not have a dominating vertex, the subgraphs induced by any two minimal connected dominating sets are isomorphic. In particular, any inclusionwise minimal connected dominating set of a connected distance-hereditary graph without dominating vertex has minimal size. In other words, connected distance-hereditary graphs without dominating vertex are connected well-dominated. Furthermore, we show that if G is a distance-hereditary graph that has a minimal connected dominating set X of size at least 2, then for any connected induced subgraph H it holds that the subgraph induced by any minimal connected dominating set of H is isomorphic to an induced subgraph of GX

Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur

Towards an Isomorphism Dichotomy for Hereditary Graph Classes

In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and algorithmic analysis of graphs. First, we develop a methodology to show isomorphism completeness of the isomorphism problem on graph classes by providing a general framework unifying various reduction techniques. Second, we generalize the concept of the modular decomposition to colored graphs, allowing for non-standard decompositions. We show that, given a suitable decomposition functor, the graph isomorphism problem reduces to checking isomorphism of colored prime graphs. Third, we extend the techniques of bounded color valence and hypergraph isomorphism on hypergraphs of bounded color size as follows. We say a colored graph has generalized color valence at most k if, after removing all vertices in color classes of size at most k, for each color class C every vertex has at most k neighbors in C or at most k non-neighbors in C. We show that isomorphism of graphs of bounded generalized color valence can be solved in polynomial time.Comment: 37 pages, 4 figure