Complexity Classification of Some Edge Modification Problems

Abstract

In an edge modiÿcation problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NP-hardness of a variety of edge modiÿcation problems with respect to some well-studied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a ÿnite set of forbidden induced subgraphs. ?

Graph Editing to a Given Neighbourhood Degree List is Fixed-Parameter Tractable

“…In fact, edge deletion problems in general tend to be NP-complete. For example, edge deletion to a cluster graph [42], bipartite graph [19], threshold graph [38], and interval graph [21] are all NP-complete. A well-studied edge insertion problem is the sandwich problem, which has applications in DNA mapping and parallel processing [23].…”

Editing to a Graph of Given Degrees

“…The edge deletion problems were considered by Yannakakis [23], Alon, Shapira and Sudakov [1]. The case when edge additions and deletions are allowed and the property is the inclusion in some hereditary graph class was considered by Natanzon, Shamir and Sharan [20] and Burzyn, Bonomo and Durán [6].…”

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“…Edge modification problems in graphs have many applications and are wellstudied in algorithmic graph theory [4,14]. The corresponding minimization problems ask to modify as few (potential) edges as possible such that an input graph is transformed into a graph with a desired property.…”