Minimal forbidden sets for degree sequence characterizations

Abstract

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.Comment: 19 pages, 4 figure

Hereditary unigraphs and Erdős–Gallai equalities

“…If π also satisfies (iii-a ′ ), then 3 = d 3 ≥ d n−2 = n − 4, so n ≤ 7. The only non-increasing, graphic sequences of length 6 or 7 satisfying these conditions are (4, 3, 3, 2, 1, 1), (4, 4, 3, 2, 2, 1), (5,4,3,3,3,1,1), and (5,5,3,3,3,2,1). The first two are split sequences, contrary to our assumption on π.…”

“…Note that n must be even, and π has as one of its realizations the complement of P 4 + (n/2 − 2)K 2 , which has the complement of P 4 + K 2 as an induced subgraph. It follows that π Rao-contains (4,4,4,4,3,3).…”

“…Theorem 6.8. The degree sequences (5, 5, 4, 4, 2, 2, 1, 1), (5, 4, 4, 2, 2, 1, 1, 1), (5, 5, 4, 2, 2, 2, 1, 1), (5, 5, 5, 4, 2, 2, 2, 1), (5, 5, 5, 5, 2, 2, 2, 2), (6, 5, 4, 4, 3, 2, 1, 1), (6, 5, 5, 4, 3, 2, 2, 1), (6, 5, 5, 4, 3, 3, 1, 1), (6, 5, 5, 5, 3, 2, 2, 2), (6,6,4,4,3,2,2,1), (6,6,5,4,3,3,2,1), (6,6,5,5,3,3,2,2), (6,6,5,5,5,3,2,2), (6,6,6,5,5,3,3,2) are elements of L 2 ; these are exactly the Rao-minimal forbidden degree sequences of split graphs.…”

“…We use Barrus and Hartke's [4] theory of degree-sequence-forcing sets to establish a forbidden induced subgraph characterization of HCU . We first define a class of graphs to be characterized by its degree sequences if for every degree sequence d, if one realization belongs to the class, then every realization belongs to the class.…”

“…If a set F is not DSF , then there exists a degree sequence d with realizations G and G ′ such that G is F-free and G ′ is not. As in [4], we call such a set an F-breaking pair. When F is finite, it is possible to identify whether it is DSF without needing to check all degree sequences.…”