Hereditary Domination in Graphs: Characterization with Forbidden Induced Subgraphs

Abstract

The leaf graph of a connected graph is obtained by joining a new vertex of degree one to each noncutting vertex. We prove that if a connected graph G is not dominated by any of its induced paths, then G is dominated by a connected induced subgraph whose leaf graph, too, is an induced subgraph of G. It follows that, for every nonempty class ${\cal D}$ of connected graphs, all of the minimal graphs not dominated by any induced subgraph isomorphic to some $D\in{\cal D}$ are cycles (of well-determined lengths) and leaf graphs of some graphs $H\notin{\cal D}$. In particular, if ${\cal D}$ is closed under the operation of taking connected induced subgraphs, then the hereditarily ${\cal D}$-dominated graphs are characterized by the following family of forbidden induced subgraphs: leaf graphs of the connected graphs that are not in ${\cal D}$, but all of their connected induced subgraphs are in ${\cal D}$, and the cycle $C_{t+2}$, where t is the length of the shortest path not in ${\cal D}$ (if ${\cal D}$ does not contain all paths). This solves a problem that was open since the 1980s. A solution for the case of induced-hereditary classes ${\cal D}$ has been found simultaneously by Bacsó by applying a different method.