A set D of vertices in a graph G = (V, E) is a weakly connected dominating set of G if D is dominating in G and the subgraph weakly induced by D is connected. The weakly connected domination number of G is the minimum cardinality of a weakly connected dominating set of G. The weakly connected domination subdivision number of a connected graph G is the minimum number of edges that must be subdiv...

We prove for every graph H there exists > 0 such that, for every graph G with |G| ≥ 2, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least |G| neighbours, or there are two disjoint sets A,B ⊆ V (G) with |A|, |B| ≥ |G| such that no edge joins A and B. It follows that for every graph H, there exists c > 0 such that for every graph G, if no induced subgraph...

Let G be a graph with p vertices and q edges an injective function, where k is positive integer. If the induced edge labeling defined by for each bijection, then f called odd Fibonacci irregular of G. A which admits graph. The irregularity strength ofes(G) minimum labeling. In this paper, some subdivision graphs obtained from vertex identification determined.

Semi-regular 4–8 meshes are refinable triangulated quadrangulations. They provide a powerful hierarchical structure for multiresolution applications. In this paper, we show how to decompose the DooSabin and Catmull-Clark subdivision schemes using 4–8 refinement. The proposed technique makes it possible to use these classical subdivision surfaces with semi-regular 4–8 meshes. Additional

We provide a characterization of several graph parameters (the acyclic chromatic number, the arrangeability, and a sequence of parameters related to the expansion of a graph) in terms of forbidden subdivisions. Let us start with several definitions. Throughout the paper, we consider only simple undirected graphs. A graph G = sdt(G) is the t-subdivision of a graph G, if G is obtained from G by r...

In this paper, we investigate how the Laplacian spectral radius changes when one graph is transferred to another graph obtained from the original graph by adding some edges, or subdivision, or removing some edges from one vertex to another. c © 2007 Elsevier Ltd. All rights reserved.

A set D of vertices of a graph G is a dominating set if every vertex in V \D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. The domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam h...

Gross and Rosen asked if the genus of a 2-dimensional complex K embeddable in some (orientable) surface is equal to the genus of the graph of appropriate barycentric subdivision of K. We answer the nonorientable genus and the Euler genus versions of Gross and Rosen's question in affirmative. We show that this is not the case for the orientable genus by proving that taking blog2 gcth barycentric...