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A graph is H-free if it does not contain an induced sub-graph isomorphic to H. We denote by P k and C k the path and the cycle on k vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for P7-free graphs, and that 5-COLORING is NP-complete for P6-free graphs. These two results improve all previous results on k-coloring Pt-free(More)
Let Pt and C denote a path on t vertices and a cycle on vertices, respectively. In this paper we study the k-coloring problem for (Pt, C)-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of P5-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle(More)
Given two graphs H1 and H2, a graph G is (H1, H2)-free if it contains no subgraph isomorphic to H1 or H2. We continue a recent study into the clique-width of (H1, H2)-free graphs and present three new classes of (H1, H2)-free graphs that have bounded clique-width. We also show the implications of our results for the computational complexity of the Colouring(More)
A graph is H-free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le and Mosca erroneously claimed that the gem and the co-gem are the only two 1-vertex P4-extensions H for which the class of H-free chordal(More)
For a positive integer k and graph G = (V, E), a k-colouring of G is a mapping c : V → {1, 2,. .. , k} such that c(u) = c(v) whenever uv ∈ E. The k-Colouring problem is to decide, for a given G, whether a k-colouring of G exists. The k-Precolouring Extension problem is to decide, for a given G = (V, E), whether a colouring of a subset of V can be extended(More)
Our study aims to contrast the neural temporal features of early stage of decision making in the context of risk and ambiguity. In monetary gambles under ambiguous or risky conditions, 12 participants were asked to make a decision to bet or not, with the event-related potentials (ERPs) recorded meantime. The proportion of choosing to bet in ambiguous(More)
A graph is H-free if it has no induced subgraph isomorphic to H. We continue a study into the boundedness of clique-width of sub-classes of perfect graphs. We identify five new classes of H-free split graphs whose clique-width is bounded. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases,(More)
A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs.(More)
We consider a committee voting setting in which each voter approves of a subset of candidates and based on the approvals, a target number of candidates are to be selected. In particular we focus on the axiomatic property called extended justified representation (EJR). Although a committee satisfying EJR is guaranteed to exist, the computational complexity(More)
We consider a committee voting setting in which each voter approves of a subset of candidates and based on the approvals, a target number of candidates are selected. Aziz et al. (2015) proposed two representation axioms called justified representation and extended justified representation. Whereas the former can be tested as well as achieved in polynomial(More)