A subgraph H of a multigraph G is called strongly spanning, if any vertex of G is not isolated in H , while it is called maximum k-edge-colorable, if H is proper k-edge-colorable and has the largest size. We introduce a graph-parameter sp(G), that coincides with the smallest k that a graph G has a strongly spanning maximum k-edge-colorable subgraph. Our first result offers some alternative defi...

A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. L. Volkmann (1998) raised the following question: Let D be a strong semicomplete multipartite digraph with a longest path of length l. Does there exist a strong spanning oriented subgraph of D with a lo...

Pedigrees are directed acyclic graphs that represent ancestral relationships between individuals in a population. Based on a schematic recombination process, we describe two simple Markov models for sequences evolving on pedigrees--Model R (recombinations without mutations) and Model RM (recombinations with mutations). For these models, we ask an identifiability question: is it possible to cons...

A subgraph H of a multigraph G is called strongly spanning, if any vertex of G is not isolated in H. H is called maximum k-edge-colorable, if H is proper k-edge-colorable and has the largest size. We introduce a graph-parameter sp(G), that coincides with the smallest k for which a multigraph G has a maximum k-edge-colorable subgraph that is strongly spanning. Our first result offers some altern...

Let G be a strongly connected directed graph. We consider the problem of computing the smallest strongly connected spanning subgraph of G that maintains the pairwise 2-vertex-connectivity of G, i.e., the 2-vertex-connected blocks of G (2VC-B). We provide linear-time approximation algorithms for this problem that achieve an approximation ratio of 6. Based on these algorithms, we show how to appr...

A graph G is supereulerian if it has a spanning eulerian subgraph. We prove that every 3-edge-connected graph with the circumference at most 11 has a spanning eulerian subgraph if and only if it is not contractible to the Petersen graph. As applications, we determine collections F1, F2 and F3 of graphs to prove each of the following (i) Every 3-connected {K1,3, Z9}-free graph is hamiltonian if ...

In the strongly connected spanning subgraph (SCSS) problem, the goal is to find a minimum weight spanning subgraph of a strongly connected directed graph that maintains the strong connectivity. In this paper, we consider the SCSS problem for two families of geometric directed graphs; t-spanners and symmetric disk graphs. Given a constant t ≥ 1, a directed graph G is a t-spanner of a set of poin...

A spanning subgraph G of a graph H is a k-detour subgraph of H if for each pair of vertices x, y ∈ V (H), the distance, distG(x, y), between x and y in G exceeds that in H by at most k. Such subgraphs sometimes also are called additive spanners. In this paper, we study k-detour subgraphs of the n-dimensional cube, Qn, with few edges or with moderate maximum degree. Let ∆(k, n) denote the minimu...

Given two nonnegative integers s and t , a graph G is (s, t)-supereulerian if for any disjoint sets X, Y ⊂ E(G) with |X | ≤ s and |Y | ≤ t , there is a spanning eulerian subgraph H of G that contains X and avoids Y . We prove that if G is connected and locally k-edge-connected, thenG is (s, t)-supereulerian, for any pair of nonnegative integers s and t with s+t ≤ k−1. We further show that if s ...

Given a complete undirected graph with non negative costs on the edges the Edge Connected Subgraph Problem consists in nding the minimum cost spanning edge connected subgraph where multi edges are allowed in the solution A lower bound for the minimum cost edge connected subgraph is obtained by solving the linear programming relaxation for this problem which coincides with the subtour relaxation...