Graph classes characterized both by forbidden subgraphs and degree sequences

Abstract

Given a set F of graphs, a graph G is F-free if G does not contain any member of F as an induced subgraph. We say that F is a degree-sequence-forcing set if, for each graph G in the class C of F-free graphs, every realization of the degree sequence of G is also in C. We give a complete characterization of the degree-sequence-forcing sets F when F has cardinality at most two.

Non-minimal Degree-Sequence-Forcing Triples

“…Let G be a degree-sequence-forcing set. By Propositions 28, 29 and 31, and Corollary 30, we have that either min{|V 1 …”

“…The authors showed in [1] that any graph that forbids any of the degree-sequence-forcing singletons or pairs other than {2K 2 , C 4 } is a unigraph. Thus any graph that forbids one of these sets and also forbids an additional graph is a unigraph.…”

“…We now apply the technique of monotone parameters, introduced in [1], to give some necessary conditions on degree-sequence-forcing sets of bipartitioned graphs.…”

“…In [1], the authors addressed the question of which classes of graphs can be characterized both in terms of a small collection of forbidden subgraphs and in terms of their degree sequences. In particular, they defined a collection F of graphs to be degree-sequence-forcing if whenever any realization of a graphic sequence π is F -free, every other realization of π is F -free as well.…”

“…In [1] the authors characterized the degree-sequence-forcing sets of order at most two. In this paper we define a degree-sequence-forcing set F to be minimal if no proper subset of F is degree-sequence-forcing.…”