Note: On maximum independent sets in P5-free graphs
Discrete Applied Mathematics
Transversals and domination in uniform hypergraphs
European Journal of Combinatorics
Note: A note on connected dominating sets of distance-hereditary graphs
Discrete Applied Mathematics
Note: On graphs for which the connected domination number is at most the total domination number
Discrete Applied Mathematics
Hypergraphs with large domination number and with edge sizes at least three
Discrete Applied Mathematics
Information Processing Letters
Maximum weight independent sets in (P6,co-banner)-free graphs
Information Processing Letters
On dominating sets whose induced subgraphs have a bounded diameter
Discrete Applied Mathematics
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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.