We consider the problem (MSSS) of finding the minimum number of arcs in a spanning strongly connected subgraph of a strongly connected digraph. This problem is NP-hard for general digraphs since it generalizes the hamiltonian cycle problem. We characterize the number of arcs in a minimum spanning strong subgraph for digraphs which are either extended semicomplete or semicomplete bipartite. Our ...

The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the (k + 2)-core of a random graph G(n, p) asymptotically almost surely has a spanning k-regular subgraph. Thus the threshold for the appearance of a k-regular subgraph of a random graph is at most the threshold for the (k + 2)-core. In particular, this pins down the point of appea...

Motivated by questions concerning optical networks, in 2003 Gargano, Hammar, Hell, Stacho, and Vaccaro defined the notions of spanning spiders and arachnoid graphs. A spider is a tree with at most one branch (vertex of degree at least 3). The spider is centred at the branch vertex (if there is any, otherwise it is centred at any of the vertices). A graph is arachnoid if it has a spanning spider...

A proper edge coloring of a graph G with colors 1, 2, 3, . . . is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graph is (3, 4)-biregular if all vertices in one part have degree 3 and all vertices in the other part have degree 4. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph G has a spanning subgraph...

We study here a problem on graphs that involves finding a subgraph of maximum node weights spanning up to k edges. We interpret the concept of “spanning” to mean that at least one endpoint of the edge is in the subgraph in which we seek to maximize the total weight of the nodes. We discuss the complexity of this problem and other related problems with different concepts of “spanning” and show t...

We introduce a generalization of the Yao graph where the cones used are adaptively centered on a set of nearest neighbors for each node, thus creating a directed or undirected spanning subgraph of a given unit disk graph (UDG). We also permit the apex of the cones to be positioned anywhere along the line segment between the node and its nearest neighbor, leading to a class of Yao-type subgraphs...

Mobile Ad hoc Networks provide flexibility and scalability which was not taken into consideration by the existing distributed systems. These networks are distributed networks and do not require any existing infrastructure. But in these types of networks there occurs some problems among which one may be occurring of fault. This may occur due to link failure, failure of nodes or network. We illus...

Given an undirected graph, finding a minimum 2-edge connected spanning subgraph is NP-hard. We solve the problem for silicate network, brother cell and sierpiński gasket rhombus.

A locally connected spanning tree of a graph G is a spanning tree T of G such that the set of all neighbors of v in T induces a connected subgraph of G for every v ∈ V (G). The purpose of this paper is to give linear-time algorithms for finding locally connected spanning trees on strongly chordal graphs and proper circular-arc graphs, respectively.

We consider three probability measures on subsets of edges of a given finite graph G, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a uniform forest, and a related conjecture is posed for a uniform connected subgraph. The former conjecture is veri...