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 i ) with 1 ≤ i < n is isometric. This kind of ordering has been introduced by Chepoi in his study on weakly modular graphs [11]. 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.

On a conjecture about the existence of isometric subgraphs

“…The best known bound which implies that an ‐vertex graph is distance preserving is , see . The graphs which satisfy this bound also have additional properties . Determining whether or not a graph is distance‐preserving is an NP‐complete problem and this makes the border bound of the conjecture more engaging.…”

“…The graphs which satisfy this bound also have additional properties . Determining whether or not a graph is distance‐preserving is an NP‐complete problem and this makes the border bound of the conjecture more engaging. For more detail and background see .…”

First-order logic axiomatization of metric graph theory

“…As shown in [52], the partial cubes admitting a distance preserving ordering are exactly the ample partial cubes that admit a corner peeling. Finally, it was proved in [76] that the problem of deciding if a graph has such a DPO is NP-complete. Proposition 1.…”

“…Since FOLB-queries can be tested in polynomial time on any graph G (see Section 8), any property that is NP-hard to recognize is unlikely to be expressible as a FOLB-query. It is known that it is NP-complete to decide whether a given graph admits a distance preserving order [76]. Therefore if P =NP, DPO cannot be expressed as a FOLB-query.…”