This study aims to determine the value of metric dimensions and local metric dimensions of relative prime graphs formed from modulo integer rings, namely . As a vertex set is and if and are ...relatively prime. By finding the pattern elements of resolving set and local resolving set, it can be shown the value of the metric dimension and the local metric dimension of graphs are and respectively, where is the number of vertices groups that formed multiple 2,3, … , and is the cardinality of set . This research can be developed by determining the value of the fractional metric dimension, local fractional metric dimension and studying the advanced properties of graphs related to their forming rings.Key Words : metric dimension; modulo ; relative prime graph; resolving set; rings.
Given a simple graph G, a set Formula omitted is a neighborhood cover set if every edge and vertex of G belongs to some Gv with Formula omitted, where Gv denotes the subgraph of G induced by the ...closed neighborhood of the vertex v. Two elements of Formula omitted are neighborhood-independent if there is no vertex Formula omitted such that both elements are in Gv. A set Formula omitted is neighborhood-independent if every pair of elements of S is neighborhood-independent. Let Formula omitted be the size of a minimum neighborhood cover set and Formula omitted of a maximum neighborhood-independent set. Lehel and Tuza defined neighborhood-perfect graphs G as those where the equality Formula omitted holds for every induced subgraph Formula omitted of G. In this work we prove forbidden induced subgraph characterizations of the class of neighborhood-perfect graphs, restricted to two superclasses of cographs: Formula omitted-tidy graphs and tree-cographs. We give as well linear-time algorithms for solving the recognition problem of neighborhood-perfect graphs and the problem of finding a minimum neighborhood cover set and a maximum neighborhood-independent set in these same classes. Finally we prove that although for complements of trees finding these optimal sets can be achieved in linear-time, for complements of bipartite graphs it is Formula omitted-hard.
Let G be a connected graph. Dominating set is a set of vertices which each vertex D has at least one neighbor in G. The minimum cardinality of D is called the domination number G(γ(G)). The metric ...dimension of G is the minimum cardinality of a series of vertices so that each vertex G is uniquely. It is determined by the distance of vector to the selected vertices. A dominating metric dimension set is a set of vertices has a dominating set D which has condition of metric dimension. The minimum cardinality is called the resolving domination number of G, (DomDim(G)). We analyze the resolving domination number of helm graph and it's operation. We study combine the existence concept of dominating set and metric dimension. We have obtained the minimum cardinality of dominating number.
Abstract
A graph is SSP (Super Strongly Perfect) if all of its (induced) subgraph H in G obsesses a (minimal dominating set) MDS that link up all of its cliques (maximal) in H. In this paper, SSP ...Structure along with its parameters (counting of maximal cliques, cardinality of dominating set (Minimal) and colourability of closed Helm graph is analysed and also Flower snark graph families are investigated.
Abstract
In broadcasting and gossiping, communication structures are modelled using graphs. In simplex model, a communication link sends messages in a particular direction. The simplex network is ...modelled by directed graphs. Certain Knodel graphs are optimal broadcast graphs. This article defines the two layer representation of Knodel digraphs and Fibonacci digraphs-both are symmetric regular bipartite digraphs. A Knodel digraph is a symmetric digraph produced by replacing each edge of an undirected Knodel graph by a symmetric pair of directed edges. A Fibonacci digraph is a symmetric digraph produced by replacing each edge of an undirected Fibonacci graph by a symmetric pair of directed edges. Decomposition of a given graph is the partitioning of edge set so that the given graph is split in to sub-structures. If all such sub-structures are isomorphic, the decomposition is isomorphic decomposition. The isomorphic decomposition of the Knodel digraphs and Fibonacci digraphs in to isomorphic paths and stars is presented. The star decomposition of both Knodel digraph and Fibonacci digraph leads to the decomposition of equi-bipartite complete digraphs in to two classes of stars.
Abstract
Domination is an important theoretic concept in graph theory. Various types of domination have been studied By
J, K
- Set domination, A subset
D
⊆
V
in a graph
G
= (
V, E
) is a
J, K
- ...Set if every Vertex
ν
∈
V
−
D
,
J
⩽ |
N
(
ν
) ∩
D
| ⩽
K
for non negative integers
J
and
K
, that is every vertex
ν
∈
V
−
D
is adjacent to at least
J
but not more than
K
Vertices in
D
. The domination number of
G
is denoted by γ
J,K
(
G
), which is the minimum cardinality of a dominating set
G
. In tis paper the
J, K
- domination number of some special graphs and its application ave been studied.
In this article, we consider the following problem proposed by Locke and Zhang in 1991: Let G be a k-connected graph with minimum degree d and X a set of m vertices on a cycle of G. For which values ...of m and k, with m >k ≥2, must G have a cycle of length at least m i n {2 d ,|V (G )|} passing through X? Fujisawa and Yamashita solved this problem for the case k ≥3 and m =k +1 in 2008. We provide an affirmative answer to this problem for the case of k ≥3 and k +1 ≤m ≤4 k +1 3 .
The degree of a vertex of a molecular graph is the number of first neighbors of this vertex. A large number of molecular-graph-based structure descriptors (topological indices) have been conceived, ...depending on vertex degrees. We summarize their main properties, and provide a critical comparative study thereof. (doi: 10.5562/cca2294) Keywords: topological index, molecular structure descriptor, vertex-degree-based topological index, molecular graph, chemical graph theory
Extracting a Hamiltonian path or cycle in a connected graph G(V,E)has a wide scope in graph theory. Determination of Hamiltonian path in different graph structures is widely researched problem. If ...h(g) is the size of a Hamiltonian walk in G then G is termed as i-Hamiltonian, if h(g)= n + i, n is the order of G. Here we characterize i- Hamiltonian-t-laceability properties in shadow graph of paths and cycles.
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