73 research outputs found

    Partitioning a graph into disjoint cliques and a triangle-free graph

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    A graph G=(V,E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques (i.e., G[A] is P_3-free) and B induces a triangle-free graph (i.e., G[B] is K_3-free). In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be NP-complete on arbitrary graphs. Here it is proved that if a graph G is bull-free, planar, perfect, K_4-free or does not contain certain holes then deciding whether G is partitionable is NP-complete. This answers an open question posed by Thomassé, Trotignon and Vušković. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs

    The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

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    International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

    Forced color classes, intersection graphs and the strong perfect graph conjecture

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    AbstractIn 1996, A. Sebő[11] raised the following two conjectures concerned with the famous Strong Perfect Graph Conjecture: (1) Suppose that a minimally imperfect graph G has a vertex p incident to 2ω(G)−2 determined edges and that its complement Ḡ has a vertex q incident to 2α(G)−2 determined edges. (An edge of G is called determined if an ω-clique of G contains both of its endpoints.) Then G is an odd hole or an odd antihole. (2) Let v0 be a vertex of a partitionable graph G. And suppose A,B to be ω-cliques of G so that v0∈A∩B. If every ω-clique K containing the vertex v0 is contained in A∪B, then G is an odd hole or an odd antihole. In this paper, we will prove (1) for a minimally imperfect graph G such that (p,q) is a determined edge of either G or Ḡ, and prove (2) for a minimally imperfect graph G such that Ḡ is C4-free and edges of Ḡ are all determined edges

    On the recognition and characterization of M-partitionable proper interval graphs

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    For a symmetric {0, 1, ⋆ }-matrix M of size m, a graph G is said to be M-partitionable, if its vertices can be partitioned into sets V1, V2, . . . , Vm, such that two parts Vi, Vj are completely adjacent if Mi,j = 1, and completely non-adjacent if Mi,j = 0 (Vi is considered completely adjacent to itself if it induces a clique, and completely non-adjacent if it induces an independent set). The complexity problem (or the recognition problem) for a matrix M asks whether the M-partition problem is polynomial-time solvable or NP-complete. The characterization problem for a matrix M asks if all M-partitionable graphs can be characterized by the absence of a finite set of forbidden induced subgraphs. These forbidden induced subgraphs are called obstructions to M. In the literature, many results were obtained by restricting the input graphs. In this thesis, we survey these results when the questions are restricted to the class of perfect graphs. We then study the recognition problem and the characterization problem when the inputs are restricted to proper interval graphs. The recognition problem can be solved by an existing algorithm, but we simplify its proof of correctness. As our main result, we prove that all the matrices of size 3 and size 4 with constant diagonal, have finitely many minimal proper interval obstructions. We also obtain partial results about matrices of arbitrary size if they have a zero diagonal

    Matrix partitions of perfect graphs

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    AbstractGiven a symmetric m by m matrix M over 0,1,*, the M-partition problem asks whether or not an input graph G can be partitioned into m parts corresponding to the rows (and columns) of M so that two distinct vertices from parts i and j (possibly with i=j) are non-adjacent if M(i,j)=0, and adjacent if M(i,j)=1. These matrix partition problems generalize graph colourings and homomorphisms, and arise frequently in the study of perfect graphs; example problems include split graphs, clique and skew cutsets, homogeneous sets, and joins.In this paper we study M-partitions restricted to perfect graphs. We identify a natural class of ‘normal’ matrices M for which M-partitionability of perfect graphs can be characterized by a finite family of forbidden induced subgraphs (and hence admits polynomial time algorithms for perfect graphs). We further classify normal matrices into two classes: for the first class, the size of the forbidden subgraphs is linear in the size of M; for the second class we only prove exponential bounds on the size of forbidden subgraphs. (We exhibit normal matrices of the second class for which linear bounds do not hold.)We present evidence that matrices M which are not normal yield badly behaved M-partition problems: there are polynomial time solvable M-partition problems that do not have finite forbidden subgraph characterizations for perfect graphs. There are M-partition problems that are NP-complete for perfect graphs. There are classes of matrices M for which even proving ‘dichotomy’ of the corresponding M-partition problems for perfect graphs—i.e., proving that these problems are all polynomial or NP-complete—is likely to be difficult

    Partitionable graphs arising from near-factorizations of finite groups

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    AbstractIn 1979, two constructions for making partitionable graphs were introduced in (by Chvátal et al. (Ann. Discrete Math. 21 (1984) 197)). The graphs produced by the second construction are called CGPW graphs. A near-factorization (A,B) of a finite group is roughly speaking a non-trivial factorization of G minus one element into two subsets A and B. Every CGPW graph with n vertices turns out to be a Cayley graph of the cyclic group Zn, with connection set (A−A)⧹{0}, for a near-factorization (A,B) of Zn. Since a counter-example to the Strong Perfect Graph Conjecture would be a partitionable graph (Padberg, Math. Programming 6 (1974) 180), any ‘new’ construction for making partitionable graphs is of interest. In this paper, we investigate the near-factorizations of finite groups in general, and their associated Cayley graphs which are all partitionable. In particular, we show that near-factorizations of the dihedral groups produce every CGPW graph of even order. We present some results about near-factorizations of finite groups which imply that a finite abelian group with a near-factorization (A,B) such that |A|⩽4 must be cyclic (already proved by De Caen et al. (Ars Combin. 29 (1990) 53)). One of these results may be used to speed up exhaustive calculations. At last, we prove that there is no counter-example to the Strong Perfect Graph Conjecture arising from near-factorizations of a finite abelian group of even order

    Borodin-Kostochka conjecture and Partitioning a graph into classes with no clique of specified size

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    For a given graph HH and the graphical properties P1,P2,…,PkP_1, P_2,\ldots,P_k, a graph HH is said to be (V1,V2,…,Vk)(V_1, V_2,\ldots,V_k)-partitionable if there exists a partition of V(H)V(H) into kk-sets V1,V2…,VkV_1, V_2\ldots,V_k, such that for each i∈[k]i\in[k], the subgraph induced by ViV_i has the property PiP_i. In 19791979, Bollob\'{a}s and Manvel showed that for a graph HH with maximum degree Δ(H)≥3\Delta(H)\geq 3 and clique number ω(H)≤Δ(H)\omega(H)\leq \Delta(H), if Δ(H)=p+q\Delta(H)= p+q, then there exists a (V1,V2)(V_1,V_2)-partition of V(H)V(H), such that Δ(H[V1])≤p\Delta(H[V_1])\leq p, Δ(H[V2])≤q\Delta(H[V_2])\leq q, H[V1]H[V_1] is (p−1)(p-1)-degenerate, and H[V2]H[V_2] is (q−1)(q-1)-degenerate. Assume that p1≥p2≥⋯≥pk≥2p_1\geq p_2\geq\cdots\geq p_k\geq 2 are kk positive integers and ∑i=1kpi=Δ(H)−1+k\sum_{i=1}^k p_i=\Delta(H)-1+k. Assume that for each i∈[k]i\in[k] the properties PiP_i means that ω(H[Vi])≤pi−1\omega(H[V_i])\leq p_i-1. Is HH a (V1,…,Vk)(V_1,\ldots,V_k)-partitionable graph? In 1977, Borodin and Kostochka conjectured that any graph HH with maximum degree Δ(H)≥9\Delta(H)\geq 9 and without KΔ(H)K_{\Delta(H)} as a subgraph, has chromatic number at most Δ(H)−1\Delta(H)-1. Reed proved that the conjecture holds whenever Δ(G)≥1014 \Delta(G) \geq 10^{14} . When p1=2p_1=2 and Δ(H)≥9\Delta(H)\geq 9, the above question is the Borodin and Kostochka conjecture. Therefore, when all pip_is are equal to 22 and Δ(H)≤8\Delta(H)\leq 8, the answer to the above question is negative. Let HH is a graph with maximum degree Δ\Delta, and clique number ω(H)\omega(H), where ω(H)≤Δ−1\omega(H)\leq \Delta-1. In this article, we intend to study this question when k≥2k\geq 2 and Δ≥13\Delta\geq 13. In particular as an analogue of the Borodin-Kostochka conjecture, for the case that Δ≥13\Delta\geq 13 and pi≥2p_i\geq 2 we prove that the above question is true
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