Let G be a minor-closed graph class. We say that is k-apex of if contains set S at most k vertices such G?S belongs to G. denote by Ak(G) the all graphs are k-apices prove every in obstruction Ak(G), i.e., minor-minimal not belonging has order 2222poly(k), where poly polynomial function whose degree depends on minor-obstructions This bound drops 22poly(k) when excludes some apex as minor.

A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a far-reaching generalization of this, conjectured by Wagner: If a class of graphs is minor-closed (i.e., it is...

The strong embedding conjecture states that every 2-connected graph has a closed 2-cell embedding in some surface , i . e . an embedding that each face is bounded by a circuit in the graph . A graph is called k -crosscap embeddable if it can be embedded in the surface of non-orientable genus k . We confirm the strong embedding conjecture for 5-crosscap embeddable graphs . As a corollary , every...

1 PROBLEM DEFINITION The theory of bidimensionality provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for a broad range of NP-hard graph problems in a broad range of graphs. This theory applies to graph problems that are " bidi-mensional " in the sense that (1) the solution value for the k ×k grid graph and similar graphs grows with k, t...

and Local Rule Learning in Attributed Networks Henry Soldano(B), Guillaume Santini, and Dominique Bouthinon 1 Université Paris 13, Sorbonne Paris Cité, L.I.P.N UMR-CNRS 7030 F-93430, Villetaneuse, France 2 Atelier de BioInformatique, ISYEB UMR 7205 CNRS MNHN UPMC EPHE, Museum d’Histoire Naturelle, F-75005, Paris, France Abstract. We address the problem of finding local patterns and related loca...

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 of each other. The Square Root problem is that of deciding whether a given graph admits a square root. This problem is only known to be NP-complete for chordal graphs and polynomial-time solvable for non-trivial minor-closed graph classes and a very l...

A geometric graph is a labeled graph whose vertices are points in the 2D plane with an isomorphism invariant under geometric transformations such as translation, rotation, and scaling. While Kuramochi and Karypis (ICDM2002) extensively studied the frequent pattern mining problem for geometric subgraphs, the maximal graph mining has not been considered so far. In this paper, we study the maximal...

A geometric graph is a labeled graph whose vertices are points in the 2D plane with isomorphism invariant under geometric transformations such as translation, rotation, and scaling. While Kuramochi and Karypis (ICDM2002) extensively studied the frequent pattern mining problem for geometric subgraphs, the maximal graph mining has not been considered so far. In this paper, we study the maximal (o...

A graph G is said to be F-residual if for every point u in G, the graph obtained by removing the closed neighborhood of u from G is isomorphic to F. Similarly, if the remove of m consecutive closed neighborhoods yields Kn, then G is called m-Kn-residual graph. Erdös determine the minimum order of the m-Kn-residual graph for all m and n, the minimum order of the connected Kn-residual graph is fo...

For several graph theoretic parameters such as vertex cover and dominating set, it is known that if their values are bounded by k then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixed-parameter algorithms on minor-closed graph classes such as planar graphs, single-crossing-minor-free graphs, and graphs of bounded gen...