Disconnected Cuts in Clawfree Graphs
Abstract
A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The decision problem whether a graph has a disconnected cut is called Disconnected Cut. This problem is closely related to several homomorphism and contraction problems, and fits in an extensive line of research on vertex cuts with additional properties. It is known that Disconnected Cut is NPhard on general graphs, while polynomialtime algorithms are known for several graph classes. However, the complexity of the problem on clawfree graphs remained an open question. Its connection to the complexity of the problem to contract a clawfree graph to the 4vertex cycle $C_4$ led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. We prove that Disconnected Cut is polynomialtime solvable on clawfree graphs, answering the question of Ito et al. The centerpiece of our result is a novel decomposition theorem for clawfree graphs of diameter 2, which we believe is of independent interest and expands the research line initiated by Chudnovsky and Seymour (JCTB 20072012) 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 novel algorithmic results, namely Disconnected Cut is polynomialtime solvable on circulararc graphs and line graphs.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.03663
 Bibcode:
 2018arXiv180303663M
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics