We give polynomial-time constant-factor approximation algorithms for the treewidth and branchwidth of any H-minor-free graph for a given graph H with crossing number at most 1. The approximation factors are 1.5 for treewidth and 2.25 for branchwidth. In particular, our result directly applies to classes of nonplanar graphs such as K5-minorfree graphs and K3,3-minor-free graphs. Along the way, w...

Localization is the first step in osteoarthritis (OA) classification. Manual classification, however, is time-consuming, tedious, and expensive. The proposed system is designed as decision support system for medical doctors to classify the severity of knee OA. A method has been proposed here to localize a joint space area for OA and then classify it in 4 steps to classify OA into KL-Grade 0, KL...

The toughness of a non-complete graph G is the minimum value of |S| ω(G−S) among all separating vertex sets S ⊂ V (G), where ω(G − S) > 2 is the number of components of G−S. It is well-known that every 3-connected planar graph has toughness greater than 1/2. Related to this property, every 3-connected planar graph has many good substructures, such as a spanning tree with maximum degree three, a...

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database...

It is proved that, if G is a K4-minor-free graph with maximum degree 3, then G is totally 4-choosable; that is, if every element (vertex or edge) of G is assigned a list of 4 colours, then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results, this shows that...

The class of all even-hole-free graphs has unbounded tree-width, as it contains complete graphs. Recently, a (even-hole, K4)-free was constructed, that still tree-width (Sintiari and Trotignon, 2019). degree arbitrarily large clique-minors. We ask whether this is necessary. prove for every graph G, if G excludes fixed H minor, then either small or wall the line induced subgraph. This can be see...

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending the tractability of many problems in dat...

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...

2 Properties excluding a planar graph 3 2.1 Path-width and tree-width . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Excluding a forest . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Planar graphs and sleeve unions . . . . . . . . . . . . . . . . . . 10 2.4 Planar graphs and grids . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Families with bounded tree-width . . . . . . . ....

We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor. In other words, we give polynomial time algorithms that, for a given H-topological-minor free graph G and a positive integer k, output an H-topological-minor free graph G′ on O(k) vertices such that G has a (connected) dominating set of size k if ...