We consider the problem of finding frequency dominators in a directed graph with a single source vertex and a single terminal vertex. A vertex x is a frequency dominator of a vertex y if and only if in each source to terminal path, the number of occurrences of x is at least equal to the number of occurrences of y. This problem was introduced in a paper by Lee et al. [11] in the context of dynam...

We prove Edmonds-Gallai type structure theorems for Mader’s edgeand vertex-disjoint paths including also capacitated variants, and state a conjecture generalizing Mader’s minimax theorems on path packings and Cunningham and Geelen’s path-matching theorem.

In this paper, we give an algorithm that, given an undirected graph G of m edges and an integer k, computes a graph G′ and an integer k′ in O(km) time such that (1) the size of the graph G′ is O(k), (2) k′ ≤ k, and (3) G has a feedback vertex set of size at most k if and only if G′ has a feedback vertex set of size at most k′. This is the first linear-time polynomial-size kernel for Feedback Ve...

Let be a connected graph with vertex set and edge . The bijective function is said to labeling of where the associated weight for If every has different weight, called an antimagic labeling. A path in vertex-labeled , two edges satisfies rainbow path. if vertices there exists Graph admits coloring, we assign each color smallest number colors induced from all weights connection denoted by In thi...

In the graph theory, two graphs are said to be isomorphic if there is a one-one, onto mapping defined between their set of vertices so as to preserving the adjacency between vertices. An isomorphism defined on a vertex set of a graph to itself is called automorphism of the given graph. Two vertices in a graph are said to be similar if there is an automorphism defined on its vertex set mapping o...

A path partition (also referred to as a linear forest) of graph G $G$ is set vertex-disjoint paths which together contain all the vertices . An isolated vertex considered be in this case. The conjecture states that every n $n$ -vertex d $d$ -regular has with at most + 1 $\frac{n}{d+1}$ paths. been proved for < 6 $d\lt 6$ We prove = $d=6$

Rainbow vertex-connection number is the minimum colors assignment to vertices of graph, such that each vertex connected by a path whose edges have distinct and denoted . The rainbow connection can be applied graphs resulting from operations. One methods create new graph perform operations between two graphs. Thus, this research uses comb product operation determine rainbow-vertex cycle complete...

Let G = (V (G), E(G)) be a finite simple graph without loops. The neighbourhood N (v) of a vertex v is the set of vertices adjacent to v. The degree d(v) of v is |N (v)|. The minimum and maximum degree of G are denoted by δ(G) and 1(G), respectively. For a vertex v ∈ V (G) and a subset S ⊆ V (G), NS(v) is the set of neighbours of v contained in S, i.e., NS(v) = N (v) ∩ S. We let dS(v) = |NS(v)|...