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.

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set is called rainbow if any two vertices in have distinct colors. The graph vertex-disconnected for x y G, there exists S such that when are nonadjacent, belong to different components $$G-S$$ ; whereas adjacent, $$S+x$$ or $$S+y$$ $$(G-xy)-S$$ . Such an x–y vertex-cut G. For vertex-disconnection number denoted ...

In 1880, P. G. Tait showed that the four colour theorem is equivalent to the assertion that every 3-regular planar graph without cut-edges is 3-edge-colourable, and in 1891, J. Petersen proved that every 3-regular graph with at most two cut-edges has a 1-factor. In this paper, we introduce the notion of collapsing all edges of a 1-factor of a 3-regular planar graph, thereby obtaining what we ca...

Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The cardinality of any smallest maximal matching in $G$ is the saturation number of $G$ and is denoted by $s(G)$. In this paper we study the saturation numbe...

In 1996, Reed proved that the domination number, γ(G), of every n-vertex graph G with minimum degree at least 3 is at most 3n/8 and conjectured that γ(H) ≤ dn/3e for every connected 3-regular (cubic) n-vertex graph H. In [1] this conjecture was disproved by presenting a connected cubic graph G on 60 vertices with γ(G) = 21 and a sequence {Gk} ∞ k=1 of connected cubic graphs with limk→∞ γ(Gk) |V...

Poljak, S. and D. Turzik, Max-cut in circulant graphs, Discrete Mathematics 108 (1992) 379-392. We study the max-cut problem in circulant graphs C,,,, where C,,, is a graph whose edge set consists of a cycle of length n and all the vertex pairs of distance r on the cycle. An efficient solution of the problem is obtained so that we show that there is always a maximum cut of a particular shape, c...

Given G = (V, E) an undirected graph and two speci ed nonadjacent nodes a and b of V , a cut separator is a subset F = δ(C) ⊆ E such that a, b ∈ V \C and a and b belong to di erent connected components of the graph induced by V \C. Given a nonnegative cost vector c ∈ R|E| + , the optimal cut separator problem is to nd a cut separator of minimum cost. This new problem is closely related to the v...