A total dominating set of a graph G with no isolated vertices is subset S the vertex such that every adjacent to in S. The domination number minimum cardinality set. In this paper, we study middle graphs. Indeed, obtain tight bounds for terms order graph. We also compute some known families graphs explicitly. Moreover, Nordhaus-Gaddum-like relations are presented

A set S of vertices of a graph G = (V, E) is a dominating set if every vertex of V (G) \ S is adjacent to some vertex in S. The domination number γ (G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ (G) is the minimum number of edges that must be subdivided in order to increase the domination number. Velammal showed that for any tree T of order at lea...

Point-based surface processing has developed into an attractive alternative to mesh-based processing techniques for a number of geometric modeling applications. By working with point cloud data directly, any processing is based on the given raw data and its underlying geometry rather than any arbitrary intermediate representations and generally artificial connectivity relations. In this paper, ...

A Roman dominating function of a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. The Roman domination subdivision number sdγR(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order t...

A growing number of Acidobacteria strains have been isolated from environments worldwide, with most isolates derived from acidic samples and affiliated with subdivision 1. We recovered 18 Acidobacteria strains from an alkaline soil, among which 11 belonged to the previously uncultured subdivision 6. Various medium formulations were tested for their effects on Acidobacteria growth.

We prove that a simple finite bipartite cubic non-planar graph contains a clean subdivision of K3.3. Here a subdivision of K3,3 is defined 1o be clean if it can be obtained from K3,3 by subdividing any edge by an even number of vertices. The proof is constructive and gives rise to a polynomial-time algorithm. @ 1998 Elsevier Science B.V. All rights reserved

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=sum d_t(G,i)$, where $d_t(G,i)$ is the numbe...