TL;DR
This paper investigates the structure of graphs with specific forced or forbidden edges based on degree sequences, revealing their properties and implications for graph connectivity and diameter.
Contribution
It characterizes the structure of forced and forbidden edge sets in degree sequences and explores their impact on graph properties like connectivity and diameter.
Findings
Diameter of such graphs is at most 3
Graphs are maximally edge-connected
Structural properties of forced and forbidden edges
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.
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