a set $s$ of vertices in a graph $g=(v,e)$ is called a total$k$-distance dominating set if every vertex in $v$ is withindistance $k$ of a vertex in $s$. a graph $g$ is total $k$-distancedomination-critical if $gamma_{t}^{k} (g - x) < gamma_{t}^{k}(g)$ for any vertex $xin v(g)$. in this paper,we investigate some results on total $k$-distance domination-critical of graphs.

Understanding the structure of a graph along with the structure of its subgraphs is important for several problems in graph theory. Two examples are the Reconstruction Conjecture and isomorph-free generation. This paper raises the question of which pairs of groups can be represented as the automorphism groups of a graph and a vertex-deleted subgraph. This, and more surprisingly the analogous qu...

Harary’s edge reconstruction conjecture states that a graph G=(V; E) with at least four edges is uniquely determined by the multiset of its edge-deleted subgraphs, i.e. the graphs of the form G − e for e∈E. It is well-known that this multiset uniquely determines the degree sequence of a graph with at least four edges. In this note we generalize this result by showing that the degree sequence of...

Let G be a graph, and k a positive integer. A fractional k-factor is a way of assigning weights to the edges of a graph G (with all weights between 0 and 1) such that for each vertex the sum of the weights of the edges incident with that vertex is k. A graph G is a fractional k-deleted graph if G − e has a fractional k-factor for each e ∈ E(G). In this paper, we obtain some sufficient condition...

This paper contains a number of estimations of the split domination number and the maximal domination number of a graph with a deleted subset of edges which induces a complete subgraph Kp. We discuss noncomplete graphs having or not having hanging vertices. In particular, for p = 2 the edge deleted graphs are considered. The motivation of these problems comes from [2] and [6], where the authors...

By Ulam’s conjecture every finite graph G can be reconstructed from its deck of vertex deleted subgraphs. The conjecture is still open, but many special cases have been settled. In particular, one can reconstruct Cartesian products. We consider the case of k-vertex deleted subgraphs of Cartesian products, and prove that one can decide whether a graph H is a kvertex deleted subgraph of a Cartesi...

We study the average probability that a discrete-time quantum walk finds marked vertex on graph. first show that, for regular graph, spectrum of transition matrix is determined by weighted adjacency an augmented then consider search distance and find formula in terms its vertex-deleted subgraph. In particular, any family approaches 1/4 as valency goes to infinity. also present more relaxed crit...

A b-coloring of a graph is proper coloring its vertices such that each color class contains vertex adjacent to at least one every other class. The b-chromatic number the largest integer k has with colors. In this work we present lower bounds for vertex-deleted subgraph graph, particularly regarding two important classes, claw-free graphs and chordal graphs. We also get G−{x}, when G large girth.

We design a new approximation polynomial-time algorithm for the graph coloring problem. Our proposal is based on selecting, in iterative manner, a critical vertex v of the graph. The criterion to select is based on choose the node with maximum degree and with maximum degree of its neighborhood into the set of vertices composing odd cycles. The algorithm consists of two embedded loops. While in ...