Testing a graph on 2-vertexand 2-edge-connectivity are two fundamental algorithmic graph problems. For both problems, different lineartime algorithms with simple implementations are known. Here, an even simpler linear-time algorithm is presented that computes a structure from which both the 2-vertexand 2-edge-connectivity of a graph can be easily “read off”. The algorithm computes all bridges a...

We propose two fixed-parameter tractable algorithms for the weighted Max-Cut problem on embedded 1-planar graphs parameterized by crossing number k of given embedding. A graph is called if it can be drawn in plane with at most one per edge. Our recursively reduce a to 3k planar graphs, using edge removal and node contraction. main algorithm then solves FCE-MaxCut introduced Liers Pardella [23]....

A 1-factor M of a cubic graph G is strong if |M ∩ T |= 1 for each 3-edge-cut T of G. It is proved in this paper that a cubic graph G has precisely three strong 1-factors if and only if the graph can be obtained from K4 via a series of 4 ↔ Y operations. Consequently, the graph G admits a Hamilton weight and is uniquely edge-3-colorable. c © 2001 Elsevier Science B.V. All rights reserved.

Testing a graph on 2-vertexand 2-edge-connectivity are two fundamental algorithmic graph problems. For both problems, different lineartime algorithms with simple implementations are known. Here, an even simpler linear-time algorithm is presented that computes a structure from which both the 2-vertexand 2-edge-connectivity of a graph can be easily “read off”. The algorithm computes all bridges a...

Poljak, S. and D. Turzik, Max-cut in circulant graphs, Discrete Mathematics 108 (1992) 379-392. We study the max-cut problem in circulant graphs C,,,, where C,,, is a graph whose edge set consists of a cycle of length n and all the vertex pairs of distance r on the cycle. An efficient solution of the problem is obtained so that we show that there is always a maximum cut of a particular shape, c...

Let G = (V,E) be a multigraph (it has multiple edges, but no loops). We call G maximally edge-connected if λ(G) = δ(G), and G super edge-connected if every minimum edge-cut is a set of edges incident with some vertex. The restricted edgeconnectivity λ′(G) of G is the minimum number of edges whose removal disconnects G into non-trivial components. If λ′(G) achieves the upper bound of restricted ...

A graph is said to be edge-superconnected if each minimum edge-cut consists of all the edges incident with some vertex of minimum degree. A graph G is said to be a {d, d + 1}semiregular graph if all its vertices have degree either d or d+1. A smallest {d, d+1}-semiregular graph G with girth g is said to be a ({d, d+1}; g)-cage. We show that every ({d, d+1}; g)-cage with odd girth g is edge-supe...

Let G 1⁄4 ðV ;EÞ be a k-regular graph with connectivity j and edge connectivity k. G is maximum connected if j 1⁄4 k, and G is maximum edge connected if k 1⁄4 k. Moreover, G is super-connected if it is a complete graph, or it is maximum connected and every minimum vertex cut is fxjðv; xÞ 2 Eg for some vertex v 2 V ; and G is super-edge-connected if it is maximum edge connected and every minimum...

We present a survey of results on the edge disjoint paths problem and re late this problem to the edge disjoint homotopic path and the edge disjoint homotopic cycle problem The latter problem is given a graph G V E embedded on a surface S and closed curves C Ck on S nd necessary and su cient conditions for the existence of pairwise edge disjoint cycles e C e Ck in G so that e Ci is homotopic to...

Characters recognition in natural images is a challenging problem, as it involves segmenting characters of various colours on various background. In this article, we present a method for segmenting images that use a colour perception graph. Our algorithm is inspired by graph cut segmentation techniques and it use an edge detection technique for filtering the graph before the graph-cut as well a...