Forced Edges and Graph Structure

Abstract

For a degree sequence, we define the set of edges that appear in every labeled realization of that sequence as forced, while the edges that appear in none as forbidden. We examine structure of graphs whose degree sequences contain either forced or forbidden edges. Among the things we show, we determine the structure of the forced or forbidden edge sets, the relationship between the sizes of forced and forbidden sets for a sequence, and the resulting structural consequences to their realizations. This includes showing that the diameter of every realization of a degree sequence containing forced or forbidden edges is no greater than 3, and that these graphs are maximally edge-connected. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

The principal Erdős--Gallai differences of a degree sequence

“…In [2,11] the author and Cloteaux independently studied forcible adjacency relationships such as these. Such a phenomenon is not restricted to split graphs like P 4 or to graphs with a nontrivial canonical decomposition; for example, in each of the nine labeled realizations of d = (4, 4, 3, 3, 3, 1), the vertices of degree 4 are adjacent, though the Erdős-Gallai difference list ∆(d) = (1, 1, 2, 2) contains no 0 term.…”