On distance-preserving elimination orderings in graphs: Complexity and algorithms

Abstract For every connected graph G , a subgraph H of G is isometric if the distance between any two vertices in H is the same in H as in G . A distance-preserving elimination ordering of G is a total ordering of its vertex-set V ( G ) , denoted ( v 1 , v 2 , … , v n ) , such that any subgraph G i = G ∖ ( v 1 , v 2 , … , v i ) with 1 ≤ i n is isometric. This kind of ordering has been introduced by Chepoi in his study on weakly modular graphs (Chepoi, 1998). We prove that it is NP-complete to decide whether such ordering exists for a given graph — even if it has diameter at most 2. Then, we prove on the positive side that the problem of computing a distance-preserving ordering when there exists one is fixed-parameter-tractable in the treewidth. Lastly, we describe a heuristic in order to compute a distance-preserving ordering when there exists one that we compare to an exact exponential time algorithm and to an ILP formulation for the problem.

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