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 $$\mathsf {FPT}$$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).
Highlights
The graph isomorphism problem (GI) is the problem of determining, given a pair of graphs G and H, whether they are isomorphic
We show that isomorphism is fixed-parameter tractable on such graphs parameterized by k with fixed degree d; in particular we give a procedure that computes in linear time a set U of vertices of size polynomial in k so that any deletion set must be found in U if one exists
We introduce a new way of parameterizing graphs by their distance to triviality, i.e. by elimination distance
Summary
The graph isomorphism problem (GI) is the problem of determining, given a pair of graphs G and H , whether they are isomorphic. It should be noted that graphs with degree bounded by d (for d ≥ 3) have unbounded tree-width and the same is a fortiori true of graphs with elimination distance k to degree d To put this parameter in context, consider the simplest notion of distance to triviality for a graph G: the number k of vertices of G that must be deleted to obtain a graph with no edges. We parameterize G by the number k of vertices that must be deleted to obtain a subgraph of G with maximum degree d This yields the parameter deletion distance to bounded degree, which we consider in Sect. It should be noted that the parameter termed generalised tree depth in [2] can be seen as a special case of elimination distance to degree 2 With this characterisation established, we are able to present the canonisation algorithm in Sect. We use the result of Bouland et al [2] on canonisation of bounded tree-depth graphs to bound the branching in the search and establish that canonisation is FPT
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
