A graph is H-free if it does not contain an induced subgraph 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 kcoloring Pt-free graphs, and almost complete the classification of complexity of k-COLORING Pt-free graphs for k ≥ 4 and t ≥ 1, leaving as the only missing case 4-COLORING P6-free graphs. We expect that 4-COLORING is polynomial time solvable for P6-free graphs; in support of this, we describe a polynomial time algorithm for 4-COLORING P6free graphs which are also P -free, where P is the graph obtained from C4 by adding a new vertex and making it adjacent to exactly one vertex on the C4.We continue the study of k-COLORING problem for P t -free graphs. This problem has been given wide attention in recent years and much progress has been made through substantial efforts by different groups of researchers [2,3,4,7,11,13,16,17,18,21,24]. We summarize these results and explain our new results below.We refer to [1] for standard graph theory terminology and [9] for terminology on computational complexity. Let G = (V, E) be a graph and H be a set of ⋆ An extended abstract of this paper has appeared in the proceedings of MFCS 2013.

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 co-gem are the only two 1-vertex P 4 -extensions H for which the class of H-free chordal graphs has bounded clique-width. In fact we prove that bull-free chordal and co-chair-free chordal graphs have clique-width at most 3 and 4, respectively. In particular, we find four new classes of H-free chordal graphs of bounded clique-width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H-free chordal graphs has bounded clique-width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of (2P 1 + P 3 , K 4 )-free graphs has bounded clique-width via a reduction to K 4 -free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique-width of H-free weakly chordal graphs.1 See also Information System on Graph Classes and their Inclusions [30], which keeps a record of graph classes for which (un)boundedness of clique-width is known. 2 This follows from results [22,33,45,57] that assume the existence of a so-called c-expression of the input graph G ∈ G combined with a result [55] that such a cexpression can be obtained in cubic time for some c ≤ 8 cw(G) − 1, where cw(G) is the clique-width of the graph G. Lemma 3 ([6]). If a prime graph G contains an induced 2P 1 + P 2 , then it contains an induced P 1 + P 4 , d-A or d-domino (see Fig. 4). Lemma 4 ([15]). If a prime graph G contains an induced subgraph isomorphic to P 1 + P 4 , then it contains one of the graphs in Figure 5 as an induced subgraph.We also use the following structural lemma due to Olariu.Lemma 5 ([54]). Every prime (bull, house)-free graph (see Fig. 6) is either K 3 -free or the complement of a 2P 2 -free bipartite graph.Lemma 32. If a prime (2P 2 , C 5 , S 1,1,2 )-free graph G contains an induced subgraph isomorphic to the net (see Fig. 5), then G is a thin spider.Proof. Suppose that G is a prime (2P 2 , C 5 , S 1,1,2 )-free graph and suppose that G contains a net, say N, with vertices a 1 , a 2 , a 3 -vertices of N), and the only edges between a 1 , a 2 , a 3 and bWe partition M as follows: For i ∈ {1, . . . , 5}, let M i be the set of vertices in M with exactly i neighbours in V (N). Let U be the set of vertices in M adjacent to every vertex of V (N). Let Z be the set of vertices in M with no neighbours in V (N). Note that Z is an independent set in G, since G is 2P 2 -free.We now analyze the structure of G through a series of claims.Claim 1. M 1 ∪ M 2 ∪ M 5 = ∅.

We consider the problem of selecting a fixed-size committee based on approval ballots. It is desirable to have a committee in which all voters are fairly represented. Aziz et al. (2015a; 2017) proposed an axiom called extended justified representation (EJR), which aims to capture this intuition; subsequently, Sanchez-Fernandez et al. (2017) proposed a weaker variant of this axiom called proportional justified representation (PJR). It was shown that it is coNP-complete to check whether a given committee provides EJR, and it was conjectured that it is hard to find a committee that provides EJR. In contrast, there are polynomial-time computable voting rules that output committees providing PJR, but the complexity of checking whether a given committee provides PJR was an open problem. In this paper, we answer open questions from prior work by showing that EJR and PJR have the same worst-case complexity: we provide two polynomial-time algorithms that output committees providing EJR, yet we show that it is coNP-complete to decide whether a given committee provides PJR. We complement the latter result by fixed-parameter tractability results.

Abstract. 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 have shown that k-colorability of P5-free graphs for k ≥ 4 does not. These authors have also shown, aided by a computer search, that 4-colorability of (P5, C5)-free graphs does have a finite forbidden induced subgraph characterization. We prove that for any k, the k-colorability of (P6, C4)-free graphs has a finite forbidden induced subgraph characterization. We provide the full lists of forbidden induced subgraphs for k = 3 and k = 4. As an application, we obtain certifying polynomial time algorithms for 3-coloring and 4-coloring (P6, C4)-free graphs. (Polynomial time algorithms have been previously obtained by Golovach, Paulusma, and Song, but those algorithms are not certifying; in fact they are not efficient in practice, as they depend on multiple use of Ramsey-type results and resulting tree decompositions of very high widths.) To complement these results we show that in most other cases the k-coloring problem for (Pt, C )-free graphs is NP-complete. Specifically, for = 5 we show that k-coloring is NP-complete for (Pt, C5)-free graphs when k ≥ 4 and t ≥ 7; for ≥ 6 we show that k-coloring is NP-complete for (Pt, C )-free graphs when k ≥ 5, t ≥ 6; and additionally, for = 7, we show that k-coloring is also NP-complete for (Pt, C7)-free graphs if k = 4 and t ≥ 9. This is the first systematic study of the complexity of the k-coloring problem for (Pt, C )-free graphs. We almost completely classify the complexity for the cases when k ≥ 4, ≥ 4, and identify the last three open cases.

No abstract