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: if G 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 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 etal. [X.Chen, A.A.McRae, L.Sun, Tree domination in graphs, Ars Combinatoria 73 (2004), pp.193203.] 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 G is a distance-hereditary graph that has a minimal connected dominating set X X of size at least 2, then for any connected induced subgraph H H it holds that the subgraph induced by any minimal connected dominating set of H H is isomorphic to an induced subgraph of G[X] G [ X ] .
connected dominating sets distance-hereditary graphs structural domination