Polynomial Kernelization for Removing Induced Claws and Diamonds

Abstract

A graph is called {claw, diamond}-free if it contains neither a claw (a K 1,3 ) nor a diamond (a K 4 with an edge removed) as an induced subgraph. Equivalently, {claw, diamond}-free graphs are characterized as line graphs of triangle-free graphs, or as linear dominoes (graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique). We consider the parameterized complexity of the {CLAW,DIAMOND}-FREE EDGE DELETION problem, where given a graph G and a parameter k, the question is whether one can remove at most k edges from G to obtain a {claw, diamond}-free graph. Our main result is that this problem admits a polynomial kernel. We complement this result by proving that, even on instances with maximum degree 6, the problem is NP-complete and cannot be solved in time 2 o(k) · |V (G)| O(1) unless the Exponential Time Hypothesis fails.

A Polynomial Kernel for Trivially Perfect Editing

“…We apply the existing technique of constructing a vertex modulator, but with a new twist: The fact that we are solving an edge modification problem enables us also to argue about the adjacency structure between the modulator and the rest of the graph, which is helpful in understanding the structure of the instance. This approach is of general nature, as witnessed by the fact that it was successfully applied to other edge modification problems as well [7,11,34].…”

“…After the announcement of this result at ESA 2015 [13], several results using the same basic technique have appeared: a quadratic vertex kernel for Threshold Editing and Chain Editing [11], a cubic vertex kernel for diamond-free Deletion [34], and a polynomial kernel for claw-diamond-free Deletion [7]. We hope that this generic methodology will find applications in other edge modification problems as well.…”

Incompressibility of H-free edge modification problems: Towards a dichotomy

“…Besides cliques and empty graphs, it is known for certain graphs H of at most 4 vertices (diamond [5,9], path [6,15,16], paw [7,13], and their complements) that H-free Edge Editing has a polynomial kernel, but these algorithms use very specific arguments exploiting the structure of H-free graphs. As there is a very deep known structure theory of claw-free (i.e, K 1,3 -free) graphs, it might be possible to obtain a polynomial kernel for Claw-free Edge Editing, but this is currently a major open question [4,10,12]. However, besides cliques and empty graphs, no H with at least 5 vertices is known where H-free Edge Editing has a polynomial kernel and there is no obvious candidate H for which one would expect a kernel.…”

An algorithm for destroying claws and diamonds

“…In the {Claw,Diamond}-Free Edge Deletion problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of edges of size at most k such that removing the edges of the set from G results a graph that does not contain an induced claw or diamond. This problem was introduced by Cygan et al [3]. The importance of this problem is due to its connection to the Claw-Free Edge Deletion problem.…”

“…The importance of this problem is due to its connection to the Claw-Free Edge Deletion problem. It is currently an open problem whether the latter problem has a polynomial kernel, and a polynomial kernel for {Claw,Diamond}-Free Edge Deletion may be a first step towards a polynomail kernel for Claw-Free Edge Deletion (see the discussion in [3]).…”