In this paper we discuss the problem of finding the treewidth and pathwidth of permutation graphs. If G[r] is a permutation graph with treewidth k, then we show that the pathwidth of G[r] is at most 2k, and we give an algo- rithm which constructs a path-decomposition with width at most 2k in time O(nk). We assume that the permutation r is given. For permutation graphs of which the treewidth is bounded by some constant, this result, together with previous results ([9], [15]), shows that the treewidth and pathwidth can be computed in linear time.

. In this paper, we study the optimum cost chromatic partition (OCCP) problem for several graph classes. The OCCP problem is the problem of coloring the vertices of a graph such that adjacent vertices get different colors and that the total coloring costs are minimum. We prove several approximation results for the OCCP problem restricted to bipartite, chordal, comparability, interval, permutation, split and unimodular graphs. We prove that there exists no polynomial approximation algorithm with ratio O(jV j 0:5 ) for the OCCP problem restricted to bipartite and interval graphs, unless P = NP . Furthermore, we propose approximation algorithms with ratio O(jV j 0:5 ) for bipartite, interval and unimodular graphs. Finally, we prove that there exists no polynomial approximation algorithm with ratio O(jV j 1 ) for the OCCP problem restricted to split, chordal, permutation and comparability graphs, unless P = NP .

The complexity of the simple maxcut problem is investigated for several special classes of graphs. It is shown that this problem is NP-complete when restricted to one of the following classes: chordal graphs, undirected path graphs, split graphs, tripartite graphs, and graphs that are the complement of a bipartite graph. The problem can be solved in polynomial time, when restricted to graphs with bounded treewidth, or cographs. We also give large classes of graphs that can be seen as generalizations of classes of graphs with bounded treewidth and of the class of the cographs, and allow polynomial time algorithms for the simple max cut problem. 1 Introduction One of the best known combinatorial graph problems is the max cut problem. In this problem, we have a weighted, undirected graph G = (V; E) and we look for a partition of the vertices of G into two disjoint sets, such that the total weight of the edges that go from one set to the other is as large as possible. In the simple max cu...

The achromatic number ψ(G) of a graph G = (V, E) is the maximum k such that V has a partition V1, V2,..., Vk into independent sets, the union of no pair of which is independent. Here we show that ψ(G) can be viewed as the maximum over all minimal elements of a partial order defined on the set of all colourings of G. We introduce a natural refinement of this partial order, giving rise to a new parameter, which we call the b-chromatic number, ϕ(G), of G. We prove that determining ϕ(G) is NP-hard for general graphs, but polynomial-time solvable for trees.

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INTRODUCTION A complete coloring of agY3fi G E# is a partition P =#V of the vertices V such that each induced subgedb #, V i P , is an independent set, and, for each pair of distinct sets V i #V j P , the induced subgub V j is not an independent set. Thelarg8W integW m for which G has a completecoloring is called the achromatic number of thegebG and is denoted by ##G#. 404 0196-6774/01 $35.00 2001 Elsevier Science All rigWW reserved The achromatic number was defined and studied by Harary et al. [7] and Harary and Hedetniemi [6].Computing the achromatic number for agG8 eral galb was proved NP-complete by Yannakakis and Gavril [11]. A simple proof of this fact appears in [5]. Bodlaender [1] proved, further, that the problem remains NP-complete even when we limit ourselves to connected gonne that are both intervalgterv and co-g3T3bV The NP-completeness of the achromatic number for trees was established only recently [9]. For gbGFT that are complements of trees thi

The achromatic number problem is to legally color the vertices of an input graph with the maximum number of colors, denoted /

A complete partition of a graph is a partition of the vertex set such that any two classes are connected by an edge. We consider the problem of finding a complete partition maximizing the number of classes. This relates to clustering into the greatest number of groups so as to minimize the diameter (inter-cluster connectivity) without concern for the intra-cluster topology. We give a randomized algorithm that approximates the complete partitioning number within a factor of O ( √ log n) on regular graphs. 1

The achromatic number problem is, given a graph G = (V, E), to find the greatest number of colors, Ψ(G), in a coloring of the vertices of G such that adjacent vertices get distinct colors and for every pair of colors some vertex of the first color and some vertex of the second color are adjacent. This problem is NP-complete even for trees. We obtain the following new results using combinatorial approaches to the problem. (1) A polynomial time O(|V | 3/8)-approximation algorithm for the problem on graphs with girth at least six. (2) A polynomial time 2-approximation algorithm for the problem on trees. This is an improvement over the best previous 7-approximation algorithm. (3) A linear time asymptotic 1.414-approximation algorithm for the problem when graph G is a tree with maximum degree d(|V |), where d: N − → N, such that d(|V |) = O(Ψ(G)). For example, d(|V |) = Θ(1) or d(|V |) = Θ(log |V |). (4) A linear time asymptotic 1.118-approximation algorithm for binary trees. We also improve the lower bound on the achromatic number of binary trees.

Polar graphs generalise bipartite, cobipartite, and split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here we show that for permutation graphs this problem can be solved in polynomial time. The result is surprising, as related problems like achromatic number and cochromatic number are NP-complete on permutation graphs. We give a polynomial-time algorithm for recognising graphs that are both permutation and polar. Prior to our result, polarity has been resolved only for chordal graphs and cographs. 1