Let ???? be a minor-closed graph class. We say that G is k -apex of if contains set S at most vertices such G\S belongs to . denote by ???? ( ) the all graphs are -apices In first paper this series, we obtained upper bounds on size in minor-obstruction ), i.e., minor-minimal not belonging ). article, provide an algorithm that, given n vertices, runs time 2 poly (k) ? 3 and either returns certif...

Mining frequent subgraphs is an interesting and important problem in the graph mining field, that from a single large has been strongly developed, recently attracted many researchers. Among them, MNI-based approaches are considered as state-of-the-art, such GraMi algorithm. Besides subgraph (FSM), closed was also developed. This practical applications fundamental premise for studies. paper prop...

A quadrangulation G on a closed surface F 2 is a simple graph embedded in F 2 so that each face of G is quadrilateral. The diagonal slide and the diagonal rotation were defined in [1] as two transformations of quadrangulations. See Fig. 1. We also call the both transformations diagonal transformations in total. If the graph obtained by a diagonal slide is not a simple graph, then we do not appl...

We define the notions tree-depth and upper chromatic number of a graph and show their relevance to local–global problems for graph partitions. In particular we show that the upper chromatic number coincides with the maximal function which can be locally demanded in a bounded coloring of any proper minor closed class of graphs. The rich interplay of these notions is applied to a solution of boun...

An axis-parallel b-dimensional box is a Cartesian product R1 ×R2 × . . .× Rb where each Ri (for 1 ≤ i ≤ b) is a closed interval of the form [ai, bi] on the real line. The boxicity of any graph G, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axis parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R1 × R2 × . . . × Rb,...

Recently a new graph convexity was introduced, arising from Steiner intervals in graphs that are a natural generalization of geodesic intervals. The Steiner tree of a set W on k vertices in a connected graph G is a tree with the smallest number of edges in G that contains all vertices of W . The Steiner interval I(W ) of W consists of all vertices in G that lie on some Steiner tree with respect...

The weak minor G of a graph G is the graph obtained from G by a sequence of edge-contraction operations on G. A weak-minor-closed family of upper embeddable graphs is a set G of upper embeddable graphs that for each graph G in G, every weak minor of G is also in G. Up to now, there are few results providing the necessary and sufficient conditions for characterizing upper embeddability of graphs...

It is known that the treewidth of a planar graph with a dominating set of size d is O( √ d) and this fact is used as the basis for several fixed parameter algorithms on planar graphs. An interesting question motivating our study is if similar bounds can be obtained for larger minor closed graph families. We say that a graph family F has the domination-treewidth property if there is some functio...