Reconfiguration of graphs with connectivity constraints
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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 CC is nested if for every C,C' ∈CC either C ∩ C' = ∅ 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 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 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 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.
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