Characterizing path graphs by forbidden induced subgraphs
Benjamin L\'ev\^eque (LGS), Fr\'ed\'eric Maffray (LGS), Myriam, Preissmann (LGS)
TL;DR
This paper characterizes path graphs by identifying all minimal forbidden induced subgraphs, providing a complete structural description of what makes a graph a path graph.
Contribution
It offers a complete list of minimal forbidden induced subgraphs that characterize path graphs, answering Renz's longstanding question from 1970.
Findings
Identified all minimal forbidden induced subgraphs for path graphs
Provided a structural characterization of path graphs
Resolved a 50-year-old open problem
Abstract
A graph is a path graph if it is the intersection graph of a family of subpaths of a tree. In 1970, Renz asked for a characterizaton of path graphs by forbidden induced subgraphs. Here we answer this question by listing all graphs that are not path graphs and are minimal with this property.
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