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P 4-Colorings and P 4-Bipartite Graphs Chinh T P_4-Colorings and P_4-Bipartite Graphs Chinh T. Hoàng, van Bang Le To cite this version: Chinh T. Hoàng, van Bang Le. P_4-Colorings and P_4-Bipartite Graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2001, 4 (2), pp.109-122. hal-00958951 HAL Id: hal-00958951 https://hal.inria.fr/hal-00958951 Submitted on 13 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science 4, 2001, 109–122 P4-Free Colorings and P4-Bipartite Graphs Ch´ınh T. Hoang` 1† and Van Bang Le2‡ 1Department of Physics and Computing, Wilfrid Laurier University, 75 University Ave. W., Waterloo, Ontario N2L 3C5, Canada 2Fachbereich Informatik, Universitat¨ Rostock, Albert-Einstein-Straße 21, D-18051 Rostock, Germany received May 19, 1999, revised November 25, 2000, accepted December 15, 2000. A vertex partition of a graph into disjoint subsets Vis is said to be a P4-free coloring if each color class Vi induces a subgraph without a chordless path on four vertices (denoted by P4). Examples of P4-free 2-colorable graphs (also called P4-bipartite graphs) include parity graphs and graphs with “few” P4s like P4-reducible and P4-sparse graphs. We prove that, given k ≥ 2, P4-FREE k-COLORABILITY is NP-complete even for comparability graphs, and for P5-free graphs. We then discuss the recognition, perfection and the Strong Perfect Graph Conjecture (SPGC) for P4-bipartite graphs with special P4-structure. In particular, we show that the SPGC is true for P4-bipartite graphs with one P3-free color class meeting every P4 at a midpoint. Keywords: Perfect graph, the Strong Perfect Graph Conjectrue, graph partition, cograph, NP-completeness 1 Introduction A graph G is perfect if, for each induced subgraph H of G, the chromatic number of H is equal to the clique number of H. Claude Berge introdued perfect graphs and conjectured around 1960’s that a graph is perfect if and only if it has no induced cycle of odd length at least five or the complement of such a cycle. Nowadays this conjecture is known as the Strong Perfect Graph Conjecture (SPGC) and is still open. We refer to [4] for more information on perfect graphs. A measure of a graph’s imperfection has been considered by Brown and Corneil [8] as follows. Given a graph G and a positive integer k, a map π : V(G) → {1,...,k} is a perfect k-coloring of G if the subgraphs induced by each color class π−1(i) is perfect. Thus, a graph is perfect if and only if it is perfect 1- colorable. Note also that, by the Perfect Graph Theorem [33], a graph G is perfect k-colorable if and only if its complement G is perfect k-colorable. In this paper we consider a particular example of perfect colorings. Our discussion is motivated by the fact that the perfection of a graph depends only on the structure of its induced paths on four vertices (denoted by P4); see [36]. In this sense, graphs with empty P4-structure (P4-free graphs) form a somewhat based graph class in discussing graph’s perfection; they are indeed perfect by a result due to Seinsche [38] (see also Jung [31]). †Research supported by NSERC. E-mail: [email protected] ‡E-mail: [email protected] 1365–8050 c 2001 Maison de l’Informatique et des Mathematiques´ Discretes` (MIMD), Paris, France 110 Ch´ınh T. Hoang` and Van Bang Le Now, we call a perfect k-coloring of a graph P4-free k-coloring if the subgraphs of that graph induced by the color classes are P4-free. Note that the P4 is self-complementary, hence G is P4-free k-colorable if and only if G is P4-free k-colorable. For general graphs, Brown [6] proved that P4-FREE k-COLORABILITY is NP-complete for k ≥ 3, and in [1], Achlioptas proved a more general result implying the NP-completeness of P4-FREE k-COLORABILITY for k ≥ 2. In the next section we shall prove that, for any integer k ≥ 2, P4-FREE k-COLORABILITY is NP-complete even for (particular) perfect graphs, and for P5-free graphs. In Section 3 we shall give some examples of P4-free 2-colorable graphs, which we also call P4-bipartite graphs. Many well understood classes of perfect graphs consists of P4-bipartite graphs only. In Sections 4 and 5, perfect P4-bipartite graphs and the SPGC for P4-bipartite graphs with special P4-structure will be discussed. The complement of a graph G is denoted by G. Graphs having no induced subgraphs isomorphic to a given graph H are called H-free. If X is a set of vertices in G, G[X] is the subgraph of G induced by X, and NG(X) is the neighborhood of X in G; that is, the set of all vertices outside X adjacent to some vertex in X. If the context is clear, we simply write N(X). The path on m vertices v1, v2,..., vm with edges vivi+1 (1 ≤ i < m) is denoted by Pm = v1v2 ···vm. The vertices v1 and vm are the endpoints of that path, the other vertices are the midpoints. The cycle on m vertices v1, v2,..., vm with edges vivi+1 (1 ≤ i < m) and v1vm is denoted by Cm = v1v2 ···vm. C2k+1 and C2k+1, k ≥ 2, are also called odd holes, respectively, odd antiholes. Graphs without odd holes and odd antiholes are called Berge graphs. 2 NP-completeness results We now consider the following problem for fixed positive integer k. P4-FREE k-COLORABILITY Is a given graph P4-free k-colorable? We show in this section that, for fixed k ≥ 2, P4-FREE k-COLORABILITY is NP-complete for perfect graphs. Notice that P4-free 1-colorability (that is, recognizing P4-free graphs) is solvable in linear time [14]. We shall reduce the following NP-complete problem ([37], see also [16]) to P4-FREE k-COLORABI- LITY. NOT-ALL-EQUAL 3SAT Given a collection C of clauses over set V of Boolean variables such that each clause has exactly three literals. Is there a truth assigment for V such that each clause in C has at least one true literal and at least one false literal? A comparability graph G is one which admits a transitive orientation G~ : If (x,y) and (y,z) are arcs of G~ , then (x,z) is also an arc of G~ . It is well known that comparability graphs are perfect. A typical example of comparability graphs are P4-free graphs, as proved by Jung [31]. Lemma 1 Given a comparability graph G, it is NP-complete to decide whether G is P4-bipartite. Proof. The problem is clearly in NP. We shall reduce NOT-ALL-EQUAL 3SAT to our problem. Let C = {C1,C2,...,Cm} be any set of clauses Ci =(ci1,ci2,ci3) given as input for NOT-ALL-EQUAL 3SAT, where the literals cik (1 ≤ i ≤ m,1 ≤ k ≤ 3) are taken from the set of variables V. We shall construct a comparability graph G which has a partition into two P4-free graphs if and only if C is satisfiable. For convenience, we call a vertex partition of a graph into two P4-free graphs a good partition of that graph. For each variable v ∈ V let G(v,v) be the graph shown in Figure 1 (left). Observation 1 G(v,v) has a good partition. Every good partition of G(v,v) must contain the labelled vertex v in one part and the labelled vertex v in the other part.✸ P4-Free Colorings and P4-Bipartite Graphs 111 For each clause Ci, let G(Ci) be the graph shown in Figure 1 (right). Observation 2 G(Ci) has a good partition. Every good partition of G(Ci) must contain two of the labelled vertices ci1,ci2,ci3 in one part and the other labelled vertex in the other part. Moreover, every partition of {ci1,ci2,ci3} into two non-empty subsets can be extended to a good partition of G(Ci). ✸ ❝ ❝.. ..❝ ❝ .... ... ✆ ❧ .... ... ✱ ❧....❝.✱ ❝ ❝ ✆ ❅ ❝ ❅ ❝.❏ ✆ ✜.. ..❝ .❝ ........❏..........✆ ❅v✜................... ❝ ... ❝ ❝ ❅❝ ... .............s ❝ ..s............ ❆ .. ❝........... ...........❝ ..❝ ✟ ❜ ✔ ❝.. ✓v❅ ✆❙ .. ◗◗❆✟s✟ ❅ ❜❜✔✧✧ ... ✓ ❙ ..❝ ❍ ✦s .❝ ❝ ❝ .. ✧ ❅ ✦ ❜ .. ❅ ❝ ✆ ❝.✧ ci1 ❍ ❝ ❅s ❝ ci3 ❜.❝ ✂❅❇ ci2 ✂❇❅ ✆ ✂ ❇❅ ❝ ❝✂ ❇❅❝ ✆❝ ❝ ❝ ❝ ❝ Fig. 1: The graphs G(v,v) (left) and G(Ci) (right) The proofs of the observations will follow by inspection, hence are omitted. We now create the graph G = G(C) from the graphs G(v,v)(v ∈ V) and the graphs G(Ci)(1 ≤ i ≤ m) as follows: For each v ∈ V and each 1 ≤ i ≤ m, connect the vertex x ∈ {v,v} in G(v,v) with the vertex cik in G(Ci) by an edge if, and only if, x is the literal cik in the clause Ci. Thus, in G, every cik (1 ≤ k ≤ 3) has exactly one neighbor outside G(Ci) which is one of the labelled vertices v,v in a graph G(v,v) (with cik ∈ {v,v} in the given NOT-ALL-EQUAL 3SAT instance). Suppose that G has a good partition into two P4-free graphs A and B. Then it is easy to see that, for all v ∈ V, if x ∈ {v,v} is adjacent to cik, then x and cik are in different parts A,B.
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