Non-minimal Degree-Sequence-Forcing Triples

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 degreesequence-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. A degreesequence-forcing set is minimal if no proper subset is degree-sequenceforcing. We characterize the non-minimal degree-sequence-forcing sets F when F has size 3.Mathematics Subject Classification (2000): 05C75, 05C07

Minimal forbidden sets for degree sequence characterizations

“…We perform a computer search for all such triples as described in Section 3, and from this search obtain a complete list of all minimal DSF triples (Theorem 3.13). In [5] the authors and M. Kumbhat characterize the non-minimal DSF triples; thus Theorem 3.13 completes the characterization of all DSF triples.…”

“…We remark that Lemmas 2.2 and 2.3 and Theorem 2.6 highlight a major difference between minimal and non-minimal DSF sets. While the numbers, orders, and edge set sizes of graphs in a minimal DSF set satisfy fairly stringent relationships, Theorem 2.8(1) and results in [5] show that this is not necessarily true for non-minimal DSF sets. For example, {2K 2 , C 4 , K n } is a non-minimal DSF triple for any n ≥ 1.…”

On 2-switches and isomorphism classes

“…Such "unigraph-producing" sets F are not the only DSF sets, but empirical evidence suggests [1,2,3] that they account for a large portion of the DSF sets containing small graphs.…”

“…Though the class of unigraphs is not hereditary (closed under taking induced subgraphs), a number of interesting hereditary graph classes, among them the threshold graphs and matrogenic graphs, contain only unigraphs. Hereditary classes of unigraphs also appear to have an important role in the study of hereditary graph classes having degree sequence characterizations (see [1,2,3]). We characterize all hereditary classes of unigraphs, including the maximum such class, in terms of their minimal forbidden induced subgraphs.…”

Hereditary unigraphs and Erdős–Gallai equalities

“…Our structural results will begin with the class of graphs containing no induced subgraph in {2K 2 , C 4 , chair}, where the chair (also known as the fork) is the tree with degree sequence (3, 2, 1, 1, 1). This class properly contains the threshold graphs and was shown in [2] to contain only unigraphs. We show that these graphs and their complements may be assembled from building blocks known as spiders by "expanding" vertices according to certain rules.…”