Minimal disconnected cuts in planar graphs

Abstract

Citation for published item:umi¡ nskiD wF nd ulusmD hF nd tewrtD eF nd hilikosD hFwF @PHISA 9winiml disonneted uts in plnr grphsF9D in pundmentls of omputtion theory X PHth snterntionl ymposiumD pg PHISD qd¡ nskD olndD eugust IUEIWD PHISD proeedingsF D ppF PRQEPSRF veture notes in omputer sieneF @WPIHAF Further information on publisher's website:Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-22177-919Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. The problem of finding a disconnected cut in a graph is NP-hard in general but polynomial-time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP-hard but its computational complexity is not known for planar graphs. We show that it is polynomial-time solvable on 3-connected planar graphs but NP-hard for 2-connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3-connected K3,3-free-minor graphs and on solving a topological minor problem in the dual. We show that the latter problem can be solved in polynomial-time even on general graphs. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP-hard for 2-connected apex graphs.

Disconnected cuts in claw-free graphs

“…The restriction to graphs of diameter 2 is natural, as every graph of diameter 1 has no disconnected and every graph of diameter at least 3 has a disconnected cut [15]. Finally, Disconnected Cut fits in the broad study of vertex cut problems with extra properties on the cut set; see [23] for an overview.…”

Disconnected Cuts in Claw-free Graphs

“…; see e.g. [24] for an overview. The above demonstrates that Disconnected Cut is of central importance to understanding many different types of problems, ranging from cut problems to homomorphism and graph contractibility problems.…”