Minimal disconnected cuts in planar graphs. Issue 4 (8th August 2016)

• Record Type: Journal Article
• Authors: Kamiński, Marcin; Paulusma, Daniël; Stewart, Anthony; Thilikos, Dimitrios M.
• Is Part Of: Networks. Volume 68:Issue 4(2016)
• Page Start: 250
• Page End: 259
• Publication Date: 2016-08-08
• Subjects: vertex cut -- connectivity -- planar graph; Network analysis (Planning) -- Periodicals; 658.4032
• DOI: 10.1002/net.21696
• Languages: English
• ISSNs: 0028-3045

Abstract

Abstract : The problem of finding a disconnected cut in a graph isNP ‐hard in general but polynomial‐time solvable on planar graphs. The problem of finding a minimal disconnected cut is alsoNP ‐hard but its computational complexity was not known for planar graphs. We show that it is polynomial‐time solvable on 3‐connected planar graphs butNP ‐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 K 3, 3 ‐free‐minor graphs and on solving a topological minor problem in the dual. In addition we show that the problem of finding a minimal connected cut of size at least 3 isNP ‐hard for 2‐connected apex graphs. Finally, we relax the notion of minimality and prove that the problem of finding a so‐called semi‐minimal disconnected cut is still polynomial‐time solvable on planar graphs. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(4), 250–259 2016