A characterization and hereditary properties for partition graphs
Discrete Mathematics
Recent examples in the theory of partition graphs
Discrete Mathematics
Journal of Graph Theory
Graph classes: a survey
Equistable series-parallel graphs
Discrete Applied Mathematics - Special issue on stability in graphs and related topics
Discrete Applied Mathematics - Special issue on stability in graphs and related topics
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Equistable distance-hereditary graphs
Discrete Applied Mathematics
On the recognition of k-equistable graphs
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
Equistable simplicial, very well-covered, and line graphs
Discrete Applied Mathematics
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In this paper we examine the connections between equistable graphs, general partition graphs and triangle graphs. While every general partition graph is equistable and every equistable graph is a triangle graph, not every triangle graph is equistable, and a conjecture due to Jim Orlin states that every equistable graph is a general partition graph. The conjecture holds within the class of chordal graphs; if true in general, it would provide a combinatorial characterization of equistable graphs. Exploiting the combinatorial features of triangle graphs and general partition graphs, we verify Orlin's conjecture for several graph classes, including AT-free graphs and various product graphs. More specifically, we obtain a complete characterization of the equistable graphs that are non-prime with respect to the Cartesian or the tensor product, and provide some necessary and sufficient conditions for the equistability of strong, lexicographic and deleted lexicographic products. We also show that the general partition graphs are not closed under the strong product, answering a question by McAvaney et al.