The edge-domsaturation number ds′(G) of a graph G = (V,E) is the least positive integer k such that every edge of G lies in an edge dominating set of cardinality k. The connected edge-domsaturation number dsc(G) of a graph G = (V,E) is the least positive integer k such that every edge of G lies in a connected edge dominating set of cardinality k. In this paper, we obtain several results connect...

Vertex Cover (VC): A vertex cover in an undirected graph G = (V,E) is a subset of vertices V ′ ⊆ V such that every edge in G has at least one endpoint in V ′. The vertex cover problem (VC) is: given an undirected graph G and an integer k, does G have a vertex cover of size k? Dominating Set (DS): A dominating set in a graph G = (V,E) is a subset of vertices V ′ such that every vertex in the gra...

Vertex Cover (VC): A vertex cover in an undirected graph G = (V,E) is a subset of vertices V ′ ⊆ V such that every edge in G has at least one endpoint in V ′. The vertex cover problem (VC) is: given an undirected graph G and an integer k, does G have a vertex cover of size k? Dominating Set (DS): A dominating set in a graph G = (V,E) is a subset of vertices V ′ such that every vertex in the gra...

A directed spanning tree in a directed graph G = (V,A) is a spanning tree such that no two arcs share their tails. In this paper, we propose an algorithm for listing all directed spanning trees of G. Its time and space complexities are O(|A| + ND(|V |, |A|)) and O(|A|+DS(|V |, |A|)), where D(|V |, |A|) and DS(|V |, |A|) are the time and space complexities of the data structure for updating the ...

Determining the minimum dominating set in connected graphs is one of most difficult problems defined as NP-hard. In this problem, it aimed to determine important nodes that can influence all via number on graph. study, an efficient near optimal algorithm showing a deterministic approach has been developed different from approximation algorithms mentioned literature for discovering set. The O(n3...

in recent years, in iran and other countries the power systems are going to move toward creating a competition structure for selling and buying electrical energy. these changes and the numerous advantages of dgs have made more incentives to use these kinds of generators than before. therefore, it is necessary to study all aspects of dgs, such as size selection and optimal placement and impact o...

The degree partition of a simple graph is its degree sequence rearranged in weakly decreasing order. Let DP(n) (respectively, DS(n)) denote the convex hull of all degree partitions (respectively, degree sequences) of simple graphs on the vertex set [n] = {1, 2, . . . , n}. We think of DS(n) as the symmetrization of DP(n) and DP(n) as the asymmetric part of DS(n). The polytope DS(n) is a well st...

Problem definition: Input: A graph represented (say) as the adjacency list for each vertex (or even just the degree of each vertex) Goal: Compute the average degree (equiv. number of edges) Concern: Seems to be impossible e.g. if all degrees ≤ 1, except possibly for a few vertices whose degree is about n. Theorem 1 [Feige, 2004]: There is an algorithm that estimates the average degree d of a co...