Disconnected Cuts in Claw-free Graphs

Barnaby Martin, Daniël Paulusma, Erik Jan Van Leeuwen, Michael Wagner
2018 European Symposium on Algorithms  
A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms exist for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to the complexity of the problem to contract a claw-free graph to the 4-vertex cycle C 4 led Ito et
more » ... al. (TCS 2011) to explicitly ask to resolve this open question. We prove that Disconnected Cut is polynomial-time solvable on claw-free graphs, answering the question of Ito et al. The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007(JCTB -2012 and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs. Graph connectivity is a crucial graph property studied in the context of network robustness. Well-studied notions of connectivity consider for example hamiltonicity, edge-disjoint spanning trees, edge cuts, vertex cuts, etc. In this paper, we study the notion of a disconnected cut, which is a vertex set U of a connected graph G such that G − U is disconnected and the subgraph G[U ] induced by U is disconnected as well. Alternatively, we say that V (G) can be partitioned into nonempty sets V 1 , V 2 , V 3 , V 4 such that no vertex of V 1 is adjacent to a vertex of V 3 (that is, V 1 is anti-complete to V 3 ) and V 2 is anti-complete to V 4 ; then both V 1 ∪ V 3 and V 2 ∪ V 4 form a disconnected cut. See Figure 1 for an example. The Disconnected Cut problem asks whether a given connected graph G has a disconnected cut.
doi:10.4230/lipics.esa.2018.61 dblp:conf/esa/MartinPL18 fatcat:znrxo6i2v5abxlkgdmb5pnrznu