Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

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Graph Coloring Problems

The maximum clique problem is a classical problem in combinatorial optimization which finds important applications in different domains. In this paper we try to give a survey of results concerning algorithms, complexity, and applications of this problem, and also provide an updated bibliography. Of course, we build upon precursory works with similar goals [39, 232, 266].

The maximum clique problem

This paper describes a general technique that can be used to obtain approximation schemes for various NP-complete problems on planar graphs. The strategy depends on decompos- ing a planar graph into subgraphs of a form we call k-outerplanar. For fixed k, the problems of interest are solvable optimally in linear time on k-outerplanar graphs by dynamic programming. For general planar graphs, if the problem is a maximization problem, such as maximum independent set, this technique gives for each k a linear time algorithm that produces a solution whose size is at least k/(k + 1)optimal. If the problem is a minimization problem, such as minimum vertex cover, it gives for each k a linear time algorithm that produces a solution whose size is at most (k + 1)/k optimal. Taking k = (c log log nl or k = (c log nl, where n is the number of nodes and c is some constant, we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approximation schemes includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominat- ing set, and minimum edge dominating set. For these and certain other problems, the proof of solvability on k-outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time.

Approximation algorithms for NP-complete problems on planar graphs

Complexity and Approximation

Easy problems for tree-decomposable graphs

A series-parallel graph can be constructed from a certain graph by recurslvely applying \"series\" and \"parallel\" connections The class of such graphs, which Is a well-known model of series-parallel electrical networks, is a subclass of planar graphs It is shown in a umfied manner that there exist hnearume algorithms for many combinatorial problems ff an input graph is restricted to the class of series-parallel graphs. These include 0) the decision problem with respect to a property characterized by a finite number of forbidden graphs, (u) the mlmmum edge (vertex) deletion problem with respect to the same property as above, and (Ul) the generalized matching problem Consequently, the following problems, among others, prove to be hnear-tlme computable for the class of series-parallel graphs. (I) the minimum vertex cover problem, (2) the maximum outerplanar (reduced) subgraph problem, (3) the minimum feedback vertex set problem, (4) the maximum (induced) hne-subgraph problem, (5) the maximum matching problem, and (6) the maximum disjoint triangle problem.

Nobuji Saito

Papers

Linear-time computability of combinatorial problems on series-parallel graphs

NP-Completeness of the Hamiltonian Cycle Problem for Bipartite Graphs

Graph Theory and Algorithms

Algorithms for routing around a rectangle

A linear 5-coloring algorithm of planar graphs