A set P of disjoint paths in a graph G is called a (complete) path cover of G if every vertex of G belongs to one of the paths in P. A path cover of any subgraph of G is called a partial path cover of G. For xed k > 0, a k-blanket of graph G is a partial path cover P of G, consisting of exactly k paths, that maximizes the size of the subgraph covered by P. A k-core of graph G is a partial path ...

We show that there are polynomial-time algorithms to compute maximum independent sets in the categorical products of two cographs and two splitgraphs. We show that the ultimate categorical independence ratio is computable in polynomial time for cographs.

Marc Hellmuth1,2∗Adrian Fritz2 Nicolas Wieseke3† Peter F. Stadler4−8 1 Department of Mathematics and Computer Science, University of Greifswald, Greifswald, Germany 2 Center for Bioinformatics, Saarland University, Saarbrücken, Germany 3 Parallel Computing and Complex Systems Group, Department of Computer Science, Leipzig University, Germany 4 Bioinformatics Group, Dept. of Computer Science & I...

Linearity and contiguity are two parameters devoted to graph encoding. Linearity is a generalisation of contiguity in the sense that every encoding achieving contiguity k induces an encoding achieving linearity k, both encoding having size Θ(k.n), where n is the number of vertices of G. In this paper, we prove that linearity is a strictly more powerful encoding than contiguity, i.e. there exist...

In this paper we introduce and investigate a new class of graphs called algebraic forests for which isomorphism testing can be done in time O(n3 log n). The class of algebraic forests admits a membership test of the same complexity, it includes cographs, trees and interval graphs, and even a joint superclass of the latter two, namely, rooted directed path graphs. In fact, our class is much larg...

Polar graphs generalise bipartite graphs, cobipartite graphs, 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 proble...

Two algorithms based upon a tree-cotree decomposition, called in this paper spanning tree technique (STT) and generalized spanning tree technique (GSTT), have been shown to be useful in computational electromagnetics. The aim of this paper is to give a rigorous description of the GSTT in terms of homology and cohomology theories, together with an analysis of its termination. In particular, the ...

We present new classes of graphs for which the isomorphism problem can be solved in polynomial time. These graphs are characterized by containing { in some local sense { only a small number of induced paths of length three. As it turns out, every such graph has a unique tree representation: the internal nodes correspond to three types of graph operations, while the leaves are basic graphs with ...

A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-conn...

A class of graphs is Kruskalian if Kruskal’s theorem on a well-quasi-ordering of finite trees provides a finite characterization in terms of forbidden induced subgraphs. Let k be a natural number. A graph is a k-cograph if its vertices can be colored with colors from the set {1, . . . ,k} such that for every nontrivial subset of vertices W there exists a partition {W1,W2} of W into nonempty sub...