A generic greedy algorithm, partially-ordered graphs and NP-completeness.

Let π be any fixed polynomial-time testable, non-trivial, hereditary property of graphs. Suppose that the vertices of a graph G are not necessarily linearly ordered but partially ordered, where we think of this partial order as a collection of (possibly exponentially many) linear orders in the natural way. We prove that the problem of deciding whether a lexicographically first maximal subgraph of G satisfying π, with respect to one of these linear orders, contains a specified vertex is NP-complete

A generic greedy algorithm, partially-ordered graphs and NP-completeness

Let π be any fixed polynomial-time testable, non-trivial, hereditary property of graphs. Suppose that the vertices of a graph G are not necessarily linearly ordered but partially ordered, where we think of this partial order as a collection of (possibly exponentially many) linear orders in the natural way. We prove that the problem of deciding whether a lexicographically first maximal subgraph of G satisfying π, with respect to one of these linear orders, contains a specified vertex is NP-complete

Special graph classes A survey

This survey attempts to describe the world of special graph classes with emphasis on special perfect graphs. The material is selected from an algorithmic point of view. For solving algorithmic problems on graphs the structure properties of graphs like descriptions by trees, algebraic generations, intersection models and others are very important. Characterizations and inclusions of classes are collected in a systematic way, the recognition complexity of classes is mentioned and a figure describes the relationships of about 120 classes w.r.t. inclusionAvailable from TIB Hannover: RO 1945(199) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLErev.ed.DEGerman

Short disjoint cycles in graphs with degree constraints

We show that each finite undirected graph G = (V,E),\vertV vertical stroke n, \vertE vertical stroke = m with minimum degree #delta#(G) #>=#,#>=#3 and maximum degree triangle = triangle (G) contains at least n:4(triangle -1)log2n pairwise vertex-disjoint cycles of length at most 4(triangle -1)log2n. Furthermore collections of such cycles can be determined within O(n"2 x m) steps. For constant triangle this means #OMEGA#(n/logn) cycles of length O(logn). This bound is also an optimum. A similar approach yields similar bounds for subgraphs with more edges than vertices instead of cycles. Futhermore also collections of many small pairwise disjoint induced subgraphs of this type can be determined within O(n"2 x m) steps similarly as for cyclesAvailable from TIB Hannover: RO 1945(214) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Powers of HHD-free graphs

The k-th power of a graph G is the graph with the same vertex set as G where two vertices are adjacent if their distance in G is at most k. In this paper we consider HHD-free graphs, i.e. the graphs where each cycle of length at least five has two chords. We show, that odd powers of HHD-free graphs are again HHD-free, and characterize those HHD-free graphs by forbidden subgraphs for which odd powers are chordal, and even powers are HHD-free or chordal respectively. (orig.)Available from TIB Hannover: RO 1945(315) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Convexity and HHD-free graphs

It is well-known that chordal graphs can be characterized via m-convexity. In this paper we introduce the notion of m"3-convexity (a relaxation of m-convexity) which is closely related to semisimplicial orderings of graphs. Using this notion we give simpler proofs of results from [8]and present some new characterizations of HHD-free graphs via m"3-convexity of disks. Moreover, we characterize weak bipolarizable graphs as the graphs for which the family of all m"3-convex sets is a convex geometry. As an application of our results we present a simple efficient criterion for deciding whether a HHD-free graph with given vertex radius function r is r-dominated by a clique. (orig.)Available from TIB Hannover: RR 1945(290) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Contact Representations of Planar Graphs - Extending a Partial Representation is Hard

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