Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

Abstract

A commonly studied means of parameterizing graph problems is the deletion distance from triviality (Guo et al., Parameterized and exact computation, Springer, Berlin, pp. 162-173, 2004), which counts vertices that need to be deleted from a graph to place it in some class for which efficient algorithms are known. In the context of graph isomorphism, we define triviality to mean a graph with maximum degree bounded by a constant, as such graph classes admit polynomial-time isomorphism tests. We generalise deletion distance to a measure we call elimination distance to triviality, based on elimination trees or tree-depth decompositions. We establish that graph canonisation, and thus graph isomorphism, is FPT when parameterized by elimination distance to bounded degree, extending results of Bouland et al. (Parameterized and exact computation, Springer, Berlin, pp. 218-230, 2012).

Vertex Deletion Parameterized by Elimination Distance and Even Less

“…(ed H ≤d ) O (1) • n O (1) and outputs a degree-d-elimination forest of depth O((ed H ≤d ) 2 ). This improves on earlier work by Bulian and Dawar [17], who gave an FPT-algorithm with an unspecified but computable parameter dependence to compute a decomposition of depth k 2 Ω(k) for k = ed H ≤d .…”

“…Our results fall in a recent line of work of using hybrid parameterizations [17,18,32,33,42,43,44,46,47,48,54] which simultaneously capture the connectivity structure of the input instance and properties of its optimal solutions. Several of these works give decomposition algorithms for parameterizations similar to ours; we discuss those which are most relevant.…”

“…The first type of graph decompositions we employ are H-elimination forests, which decompose graphs of bounded H-elimination distance. The H-elimination distance ed H (G) of an undirected graph G is a graph parameter that was recently introduced [17,18,57], which admits a recursive definition similar to treedepth. If G is connected and belongs to H, then ed H (G) = 0.…”

A Compound Logic for Modification Problems: Big Kingdoms Fall from Within

“…The development of parameterized algorithms for cut problems is a popular trend in parameterized algorithms [17,34,70,81,83,97,99]. More involved modification measures of vertex set removals, related to treewidth or treedepth, have been considered very recently [4,19,20,41,73].…”

“…In a recent wave of results, alternative quality measures of the "modulator" to some graph class were considered, other than just its size [3,5,16,19,20,41,49,73,95]. The first step in this direction was done by Bulian and Dawar [19,20] who considered the elimination distance to some graph class G. We assume that the target graph class is G = Mod(γ 0 ), for some γ 0 ∈ CMSOL. Also, for k ≥ 1, let γ k = β γ k−1 , where β demands than torso(G, X) is edgeless.…”