Connected max cut is polynomial for graphs without the excluded minor $$K_5\backslash e$$ K 5 \ e

Brahim Chaourar ()
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Brahim Chaourar: Imam Mohammad Ibn Saud Islamic University (IMSIU)

Journal of Combinatorial Optimization, 2020, vol. 40, issue 4, No 2, 869-875

Abstract: Abstract Given a graph $$G=(V, E)$$ G = ( V , E ) , a connected cut $$\delta (U)$$ δ ( U ) is the set of edges of E linking all vertices of U to all vertices of $$V\backslash U$$ V \ U such that the induced subgraphs G[U] and $$G[V\backslash U]$$ G [ V \ U ] are connected. Given a positive weight function w defined on E, the connected maximum cut problem (CMAX CUT) is to find a connected cut $$\varOmega $$ Ω such that $$w(\varOmega )$$ w ( Ω ) is maximum among all connected cuts. CMAX CUT is NP-hard even for planar graphs, and thus for graph without the excluded minor $$K_5$$ K 5 . In this paper, we prove that CMAX CUT is polynomial for the class of graphs without the excluded minor $$K_5\backslash e$$ K 5 \ e , denoted by $${\mathcal {G}}(K_5\backslash e)$$ G ( K 5 \ e ) . We deduce two quadratic time algorithms: one for the minimum cut problem in $${\mathcal {G}}(K_5\backslash e)$$ G ( K 5 \ e ) without computing the maximum flow, and another one for Hamilton cycle problem in the class of graphs without the two excluded minors the prism $$P_6$$ P 6 and $$K_{3, 3}$$ K 3 , 3 . This latter problem is NP-complete for maximal planar graphs.

Keywords: Max cut; Connected max cut; Polynomial algorithm; Min cut; Graphs without the excluded minor $$K_5\backslash e$$ K 5 \ e; Hamilton cycle problem; 90C27 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10878-020-00637-6

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