Minimal forbidden sets for degree sequence characterizations

• Published in: arXiv.org
• Main Authors: Barrus, Michael D, Hartke, Stephen G
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 03.10.2013
• Subjects: Graph theory; Graphs; Mathematics - Combinatorics
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1310.1109

Given a set \(\mathcal{F}\) of graphs, a graph \(G\) is \(\mathcal{F}\)-free if \(G\) does not contain any member of \(\mathcal{F}\) as an induced subgraph. Barrus, Kumbhat, and Hartke [M. D. Barrus, M. Kumbhat, and S. G. Hartke, Graph classes characterized both by forbidden subgraphs and degree sequences, J. Graph Theory (2008), no. 2, 131--148] called \(\mathcal{F}\) a degree-sequence-forcing (DSF) set if, for each graph \(G\) in the class \(\mathcal{C}\) of \(\mathcal{F}\)-free graphs, every realization of the degree sequence of \(G\) is also in \(\mathcal{C}\). A DSF set is minimal if no proper subset is also DSF. In this paper, we present new properties of minimal DSF sets, including that every graph is in a minimal DSF set and that there are only finitely many DSF sets of cardinality \(k\). Using these properties and a computer search, we characterize the minimal DSF triples.