The weakly connected domination subdivision number sdγw(G) of a connected graph G is the minimum number of edges which must be subdivided (where each edge can be subdivided at most once) in order to increase the weakly connected domination number. The graph is strongγw-subdivisible if for each edge uv ∈ E(G) we have γw(Guv) > γw(G), where Guv is a graph G with subdivided edge uv. The graph is s...

A graph G = (V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if (x, y) ∈ E. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A face subdivision of a triangular grid graph is replacing some of its cells by plane copies of the...

The locating-chromatic number of a graph combines two concepts, namely coloring vertices and partition dimension graph. is the smallest k such that G has locating k-coloring, denoted by χL(G). This article proposes procedure for obtaining an origami its subdivision (one vertex on outer edge) through theorems with proofs.

This paper presents a phenomenon which sometimes occurs in tetravalent bipartite locally dart-transitive graphs, called Base Graph–Connection Graph dissection. In this dissection, each white vertex is split into two vertices of valence 2 so that the connected components result are isomorphic. Given whose subdivision isomorphic to component, and Connection Graph, describes how overlap, we can, s...

Let G be a Berge graph such that no induced subgraph is a 4-cycle or a line-graph of a bipartite subdivision of K4. We show that every such graph G either is a complete graph or has an even pair.

Let G be a Berge graph such that no induced subgraph is a 4-cycle or a line-graph of a bipartite subdivision of K4. We show that every such graph G either is a complete graph or has an even pair.

A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-conn...

Recently, the subdivision Randic index was introduced. In this paper, we present new version of Randic index by using some graph operator and in related to the subdivision Randic index. Next, by using some results about this version, it is computed for TUC C (R) nanotubes. 4 8

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