Journal ArticleDOI
Elimination graphs
Yuli Ye,Allan Borodin +1 more
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It turns out that if the authors take <i>P</i> as a graph with maximum independent set size no greater than <i*k</i>, then this definition gives a natural generalization of both chordal graphs and (<i-k + 1)-claw-free graphs.Abstract:
In this article we study graphs with inductive neighborhood properties. Let P be a graph property, a graph G = (V, E) with n vertices is said to have an inductive neighborhood property with respect to P if there is an ordering of vertices v1, …, vn such that the property P holds on the induced subgraph G[N(vi)∩ Vi], where N(vi) is the neighborhood of vi and Vi = {vi, …, vn}. It turns out that if we take P as a graph with maximum independent set size no greater than k, then this definition gives a natural generalization of both chordal graphs and (k + 1)-claw-free graphs. We refer to such graphs as inductive k-independent graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are inductive k-independent for small k. In particular, any intersection graph of translates of a convex object in a two dimensional plane is an inductive 3-independent graph; furthermore, any planar graph is an inductive 3-independent graph. For any fixed constant k, we develop simple, polynomial time approximation algorithms for inductive k-independent graphs with respect to several well-studied NP-complete problems. Our generalized formulation unifies and extends several previously known results.read more
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References
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The ellipsoid method and its consequences in combinatorial optimization
TL;DR: The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.
Book ChapterDOI
Paths, Trees, and Flowers
TL;DR: A graph G is a set of elements called vertices and a finite set of edges called edges such that each edge meets exactly two vertices, called the end-points of the edge as mentioned in this paper.
Book
Graph Coloring Problems
Tommy R. Jensen,Bjarne Toft +1 more
TL;DR: In this article, the Conjectures of Hadwiger and Hajos are used to define graph types, such as planar graph, graph on higher surfaces, and critical graph.
Journal ArticleDOI
Approximation algorithms for NP-complete problems on planar graphs
TL;DR: A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set.
Journal ArticleDOI
Matroids and the greedy algorithm
TL;DR: Linear-algebra rank is the solution to an especially tractable optimization problem which are linear programs relative to certain derived polyhedra.