We prove a conjecture stating that the branchwidth of a graph and the branchwidth of the graph’s cycle matroid are equal if the graph has a cycle of length at least 2.

We prove a conjecture of Saito that if a graph G with ≥ 3 has no cycle of length 1 (mod 3), then G has an induced subgraph which is isomorphic to the Petersen graph. The above result strengthened the result by Dean et al. that every 2-connected graph with ≥ 3 has a (1 mod 3)-cycle if G is not isomorphic to the Petersen graph. ? 2001 Elsevier Science B.V. All rights reserved.

A signed graph is a graph whose edges are given ±1 weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal ±1 matrix. For a signed graph Σ on n vertices, its exterior kth power, where k = 1, . . . , n−1, is a graph ∧k Σ w...

In the paper graphs are assumed to be finite, undirected and without loops, though they may contain multiple edges. We will also consider pseudo-graphs, which, in contrast with graphs, may contain loops. Thus graphs are pseudo-graphs. We accept the convention that a loop contributes to the degree of a vertex by two. The set of vertices and edges of a pseudo-graph G will be denoted by V (G) and ...

Fine-grained reductions have established equivalences between many core problems with Õ(n3)-time algorithms on n-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also have Õ(mn)-time algorithms on m-edge n-node weighted graphs, and such algorithms have wider applicability. Are these mn boun...

Results related to integer flows and cycle covers are presented. A cycle cover of a graph G is a collection %Y of cycles of G which covers all edges of G; U is called a cycle m-cover of G if each edge of G is covered exactly m times by the members of V. By using Seymour’s nowhere-zero 6-flow theorem, we prove that every bridgeless graph has a cycle 6-cover associated to covering of the edges by...

A planar 3-connected graph G is essentially 4-connected if for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. B. Jackson and N.C. Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that |V (C)| ≥ 2n+4 5 . For a cubic essentially 4-connected planar graph G, B. Grünbaum, J. Malkevitch , and C.-Q. ...

The notion of a p-competition graph and the p-competition number of a graph were introduced by S. -R. Kim, T. A. McKee, F. R. McMorris, and F. S. Roberts as a generalization of a competition graph and the competition number of a graph, respectively. Let p be a positive integer. The p-competition graph Cp(D) of a digraph D = (V, A) is a (simple undirected) graph which has the same vertex set V a...

A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H . In this paper, we give a new sufficient condition for a weighted graph to...