Shifting paths to avoidable ones

Abstract

An extension of an induced path P in a graph G is an induced path P false′ such that deleting the endpoints of P false′ results in P. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced cycle. In 2019, Beisegel, Chudovsky, Gurvich, Milanič, and Servatius conjectured that every graph that contains an induced k‐vertex path also contains an avoidable induced path of the same length, and proved the result for k = 2. The case k = 1 was known much earlier, due to a work of Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all k in 2020 by Bonamy, Defrain, Hatzel, and Thiebaut. In the present paper, using a similar approach, we strengthen their result from a reconfiguration point of view. Namely, we show that in every graph, each induced path can be transformed to an avoidable one by a sequence of shifts, where two induced k‐vertex paths are shifts of each other if their union is an induced path with k + 1 vertices. We also obtain analogous results for not necessarily induced paths and for walks. In contrast, the statement cannot be extended to trails or to isometric paths.

Computing and Listing Avoidable Vertices and Paths

“…Therefore our recognition algorithm for testing whether a given induced path is avoidable takes O(n • m) time. As a side remark of the later algorithm, we partially resolve an open question raised in [10].…”

“…For instance, one could compute a minimum length of a sequence of shifts transforming an induced path P k to an avoidable induced path. Gurvich et al [10] proved that each induced path can be transformed to an avoidable one by a sequence of shifts, where two induced paths on k vertices are shifts of each other if their union is an induced path on k + 1 vertices. To compute efficiently a minimum length of shifts, one could construct a graph H that encodes all neighboring induced paths on k vertices of G. In particular, the nodes of H correspond to all induced paths on k vertices in G and two nodes in H are adjacent if and only if their union is an induced path on k + 1 vertices in G. Note that H contains O(n k ) nodes and can be constructed in n O(k) time.…”

“…Now given an induced path P k on k vertices in G we may ask the shortest path in H from the node that corresponds to P k towards a marked node that corresponds to an avoidable path. Such a path always exists from the results of [10] and can be computed in time linear in the size of H . Therefore, for fixed k, our algorithm computes a minimum length of sequence of shifts in polynomial time answering an open question given in [10].…”

“…Regarding the existence of an avoidable induced path of arbitrary length, Bonamy et al [9] settled a conjecture in [8] and showed that every graph is either P k -free or contains an avoidable P k . Gurvich et al [10] strengthened the later result by showing that every induced path can be shifted in an avoidable path, in the sense that there is a sequence of neighboring induced paths of the same length. Although the provided proof in [10] is constructive and identifies an avoidable path given an induced path, the proposed algorithm was not settled whether it runs in polynomial time.…”

“…Gurvich et al [10] strengthened the later result by showing that every induced path can be shifted in an avoidable path, in the sense that there is a sequence of neighboring induced paths of the same length. Although the provided proof in [10] is constructive and identifies an avoidable path given an induced path, the proposed algorithm was not settled whether it runs in polynomial time.…”