Hierarchical complexity of 2-clique-colouring weakly chordal graphs and perfect graphs having cliques of size at least 3

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• 作者：Hé ; lio B. Macê ; do Filho^a ; ^{helio@cos.ufrj.br" class="auth_mail" title="E-mail the corresponding author} ; Raphael C.S. Machado^b ; ^{rcmachado@inmetro.gov.br" class="auth_mail" title="E-mail the corresponding author} ; Celina M.H. de Figueiredo^a ; ^{celina@cos.ufrj.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Clique-colouring ; Hierarchical complexity ; Perfect graphs ; Weakly chordal graphs ; (alpha beta)-Polar graphs
• 刊名：Theoretical Computer Science
• 出版年：2016
• 出版时间：7 March 2016
• 年：2016
• 卷：618
• 期：Complete
• 页码：122-134
• 全文大小：874 K

文摘

A clique of a graph is a maximal set of vertices of size at least 2 that induces a complete graph. A k-clique-colouring of a graph is a colouring of the vertices with at most k   colours such that no clique is monochromatic. Défossez proved that the 2-clique-colouring of perfect graphs is a 111111111&_pii=S0304397516000517&_rdoc=1&_issn=03043975&md5=3f3e877536aae9f3f21fe8967ef2d9c6">${\mathrm{\Sigma }}_2^P$-complete problem (Défossez (2009) [4]). We strengthen this result by showing that it is still 111111111&_pii=S0304397516000517&_rdoc=1&_issn=03043975&md5=3f3e877536aae9f3f21fe8967ef2d9c6">${\mathrm{\Sigma }}_2^P$-complete for weakly chordal graphs. We then determine a hierarchy of nested subclasses of weakly chordal graphs whereby each graph class is in a distinct complexity class, namely 111111111&_pii=S0304397516000517&_rdoc=1&_issn=03043975&md5=3f3e877536aae9f3f21fe8967ef2d9c6">${\mathrm{\Sigma }}_2^P$-complete, 111111111&_pii=S0304397516000517&_rdoc=1&_issn=03043975&md5=9516dcfe2e560ca6f233ec9eb3eeda97" title="Click to view the MathML source">NP$\mathcal{NP}$-complete, and 111111111&_pii=S0304397516000517&_rdoc=1&_issn=03043975&md5=c416288028be04c346c261c0790a82ad" title="Click to view the MathML source">P$\mathcal{P}$. We solve an open problem posed by Kratochvíl and Tuza to determine the complexity of 2-clique-colouring of perfect graphs with all cliques having size at least 3 (Kratochvíl and Tuza (2002) [7]), proving that it is a 111111111&_pii=S0304397516000517&_rdoc=1&_issn=03043975&md5=3f3e877536aae9f3f21fe8967ef2d9c6">${\mathrm{\Sigma }}_2^P$-complete problem. We then determine a hierarchy of nested subclasses of perfect graphs with all cliques having size at least 3 whereby each graph class is in a distinct complexity class, namely 111111111&_pii=S0304397516000517&_rdoc=1&_issn=03043975&md5=3f3e877536aae9f3f21fe8967ef2d9c6">${\mathrm{\Sigma }}_2^P$-complete, 111111111&_pii=S0304397516000517&_rdoc=1&_issn=03043975&md5=9516dcfe2e560ca6f233ec9eb3eeda97" title="Click to view the MathML source">NP$\mathcal{NP}$-complete, and 111111111&_pii=S0304397516000517&_rdoc=1&_issn=03043975&md5=c416288028be04c346c261c0790a82ad" title="Click to view the MathML source">P$\mathcal{P}$.