We consider connectivity-augmentation problems in a setting where each potential new edge has a nonnegative cost associated with it, and the task is to achieve a certain connectivity target with at most p new edges of minimum total cost. The main result is that the minimum cost augmentation of edge-connectivity from k − 1 to k with at most p new edges is fixed-parameter tractable parameterized ...

Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are the classical results of Tutte and Nash-Williams from 1961 which show that a λ...

In the edge connectivity augmentation problem one wants to find an edge set of minimum total capacity that increases the edge connectivity of a given undirected graph by τ . It is a known non-trivial property of the edge connectivity augmentation problem that there is a sequence of edge sets E1, E2, . . . , such that ⋃ i≤τ Ei optmially increases the connectivity by τ , for any integer τ . The m...

In this paper, we develop a polynomial time algorithm to find out all the minilnum cyclic edge cutsets of a 3-regular graph, and therefore to determine the cyclic edge connectivity of a cubic graph. The algorithm is recursive, with complexity bounded by O(n31og2 n). The algorithm shows that the number of mini~um cyclic edge cut sets of a 3-regular graph G is polynornial in v( G) and that the mi...

Global connectivity is a quite important issue for networks. The failures of some key edges may lead to breakdown of the whole system. How to find them will provide a better understanding on system robustness. Based on topological information, we propose an approach named LE (link entropy) to quantify the edge significance on maintaining global connectivity. Then we compare the LE with the othe...

The aim of edge editing or modification problems is to change a given graph by adding and deleting of a small number of edges in order to satisfy a certain property. We consider the Edge Editing to a Connected Graph of Given Degrees problem that for a given graph G, non-negative integers d, k and a function δ : V (G) → {1, . . . , d}, asks whether it is possible to obtain a connected graph G′ f...

In this paper, we study the relationship between eigenvalues and the existence of certain subgraphs in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a t-edge-connected d-regular graph, when t ≤ d − 2. This work extends some classical results of von Baebler and Berge and more recent work of Cioabă, Gregory, and Haemers. ...

Let c be a proper k-coloring of a connected graph G and Π = (V1, V2, . . . , Vk) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple cΠ(v) := (d(v, V1), d(v, V2), . . . , d(v, Vk)), where d(v, Vi) = min{d(v, x) | x ∈ Vi}, 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then ...

Let G be a graph. The core of G, denoted by G∆, is the subgraph of G induced by the vertices of degree ∆(G), where ∆(G) denotes the maximum degree of G. A k-edge coloring of G is a function f : E(G)→ L such that |L| = k and f(e1) 6= f(e2) for all two adjacent edges e1 and e2 of G. The chromatic index of G, denoted by χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is...

Let l > 0 and k ≥ 0 be two integers. Denote by C(l, k) the family of 2-edge-connected graphs such that a graph G ∈ C(l, k) if and only if for every bond S ⊂ E(G) with |S| ≤ 3, each component of G − S has order at least (|V (G)| − k)/l. In this paper we prove that if a 3-edge-connected graph G ∈ C(12, 1), then G is supereulerian if and only if G cannot be contracted to the Petersen graph. Our re...