Reconfiguration of graphs with connectivity constraints

TL;DR

This paper studies the reconfiguration of graphs with degree sequences under nested connectivity constraints, providing polynomial algorithms for existence and transformation sequences.

Contribution

It introduces a polynomial-time method to determine the existence of such graphs and to transform one into another via flips while maintaining constraints.

Findings

Decides in polynomial time if a graph with given degree sequence and nested connectivity exists.

Provides a polynomial-time algorithm to transform one such graph into another via flips.

Ensures all intermediate graphs also satisfy the degree and connectivity constraints.

Abstract

A graph $G$ realizes the degree sequence $S$ if the degrees of its vertices is $S$. Hakimi gave a necessary and sufficient condition to guarantee that there exists a connected multigraph realizing $S$. Taylor later proved that any connected multigraph can be transformed into any other via a sequence of flips (maintaining connectivity at any step). A flip consists in replacing two edges $ab$ and $cd$ by the diagonals $ac$ and $bd$. In this paper, we study a generalization of this problem. A set of subsets of vertices $\mathcal{CC}$ is \emph{nested} if for every $C,C' \in \mathcal{CC}$ either $C \cap C' = \emptyset$ or one is included in the other. We are interested in multigraphs realizing a degree sequence $S$ and such that all the sets of a nested collection $\mathcal{CC}$ induce connected subgraphs. Such constraints naturally appear in tandem mass spectrometry. We show that it is possible to decide in polynomial if there exists a graph realizing $S$ where all the sets in $\mathcal{CC}$ induce connected subgraphs. Moreover, we prove that all such graphs can be obtained via a sequence of flips such that all the intermediate graphs also realize $S$ and where all the sets of $\mathcal{CC}$ induce connected subgraphs. Our proof is algorithmic and provides a polynomial time approximation algorithm on the shortest sequence of flips between two graphs whose ratio depends on the depth of the nested partition.