Note: A note on connected dominating sets of distance-hereditary graphs

• Authors: Oliver Schaudt
• Venue: Discrete Applied Mathematics
• Year: 2012

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: 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. This answers a question of Chen et al. [X. Chen, A.A. McRae, L. Sun, Tree domination in graphs, Ars Combinatoria 73 (2004), pp. 193-203.] asking for non-trivial graph classes where the minimal size of a connected dominating set inducing a tree can be computed efficiently. 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].