On 2-switches and isomorphism classes

Abstract

A 2-switch is an edge addition/deletion operation that changes adjacencies in the graph while preserving the degree of each vertex. A well known result states that graphs with the same degree sequence may be changed into each other via sequences of 2-switches. We show that if a 2-switch changes the isomorphism class of a graph, then it must take place in one of four configurations. We also present a sufficient condition for a 2-switch to change the isomorphism class of a graph. As consequences, we give a new characterization of matrogenic graphs and determine the largest hereditary graph family whose members are all the unique realizations (up to isomorphism) of their respective degree sequences.Comment: 11 pages, 6 figure

Minimal forbidden sets for degree sequence characterizations

“…In [2] the first author established a criterion for recognizing such sets F. The graphs R, R, S, and S are shown in Figure 4. The join of two graphs is indicated with the symbol ∨.…”

On Fractional Realizations of Graph Degree Sequences

“…Since H is not split, it must contain 2K 2 or C 4 as an induced subgraph. Assume that H induces 2K 2 , and let ab and cd be the edges of an induced copy of 2K 2 For k ∈ {1, 3}, let N k denote the set of vertices of H − C that have exactly k neighbors in C.…”

Neighborhood degree lists of graphs

“…They determined which sequences of integer pairs can be realized by a graph, pseudograph, or multigraph, and later Das [6] characterized the integer-pair sequences that correspond to a single graph (up to isomorphism). Figure 1: A graphical depiction of ((2, 2, 1), (3,2), (3,2), (2,2), (3)).…”

“…, τ n dn )) whose elements are lists of the degrees in G of the neighbors of a given vertex. For example, if G is the graph obtained by attaching a pendant vertex to a chordless 4-cycle, then τ (G) = ((2, 2, 1), (3,2), (3,2), (2,2), (3)). Notice how the degree sequence (3, 2, 2, 2, 1) of G is apparent from the lengths of the elements of τ (G) (we call these elements the component lists).…”

“…In situations where it becomes necessary to distinguish between nonisomorphic graphs having the same degree sequence, it may be possible to do so by consulting their NDLs. The first author used NDLs in this way in a proof in [2] (see Theorem 2.3 therein).…”