Vertex Identification in Grids and Prisms

A red-white coloring of a nontrivial connected graph [Formula: see text] of diameter [Formula: see text] is an assignment of red and white colors to the vertices of [Formula: see text] where at least one vertex is colored red. Associated with each vertex [Formula: see text] of [Formula: see text] is a [Formula: see text]-vector, called the code of [Formula: see text], whose [Formula: see text]th coordinate is the number of red vertices at distance [Formula: see text] from [Formula: see text]. A red-white coloring of [Formula: see text] for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of [Formula: see text]. A graph [Formula: see text] possessing an ID-coloring is an ID-graph. The minimum number of red vertices among all ID-colorings of an ID-graph [Formula: see text] is the identification number or ID-number of [Formula: see text]. It is shown that the grid [Formula: see text] is an ID-graph if and only [Formula: see text] and the prism [Formula: see text] is an ID-graph if and only if [Formula: see text].

Distance Vertex Identification in Graphs

Journal of Interconnection Networks ◽

10.1142/s0219265921500055 ◽

2021 ◽

pp. 2150005

Author(s):

Gary Chartrand ◽

Yuya Kono ◽

Ping Zhang

Keyword(s):

Positive Integer ◽

Connected Graph ◽

Identification Number ◽

Minimum Number ◽

Positive Integers ◽

A red-white coloring of a nontrivial connected graph [Formula: see text] is an assignment of red and white colors to the vertices of [Formula: see text] where at least one vertex is colored red. Associated with each vertex [Formula: see text] of [Formula: see text] is a [Formula: see text]-vector, called the code of [Formula: see text], where [Formula: see text] is the diameter of [Formula: see text] and the [Formula: see text]th coordinate of the code is the number of red vertices at distance [Formula: see text] from [Formula: see text]. A red-white coloring of [Formula: see text] for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of [Formula: see text]. A graph [Formula: see text] possessing an ID-coloring is an ID-graph. The problem of determining those graphs that are ID-graphs is investigated. The minimum number of red vertices among all ID-colorings of an ID-graph [Formula: see text] is the identification number or ID-number of [Formula: see text] and is denoted by [Formula: see text]. It is shown that (1) a nontrivial connected graph [Formula: see text] has ID-number 1 if and only if [Formula: see text] is a path, (2) the path of order 3 is the only connected graph of diameter 2 that is an ID-graph, and (3) every positive integer [Formula: see text] different from 2 can be realized as the ID-number of some connected graph. The identification spectrum of an ID-graph [Formula: see text] is the set of all positive integers [Formula: see text] such that [Formula: see text] has an ID-coloring with exactly [Formula: see text] red vertices. Identification spectra are determined for paths and cycles.

On the metric dimension and diameter of circulant graphs with three jumps

10.1142/s1793830918500088 ◽

2018 ◽

Vol 10 (01) ◽

pp. 1850008

Author(s):

Muhammad Imran ◽

A. Q. Baig ◽

Saima Rashid ◽

Andrea Semaničová-Feňovčíková

Keyword(s):

Positive Integer ◽

Connected Graph ◽

Metric Spaces ◽

Metric Dimension ◽

Resolving Set ◽

Circulant Graphs ◽

Infinite Class ◽

Metric Properties ◽

Minimum Number ◽

Metric Basis

Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.

The minimum eccentric distance sum of trees with given distance k-domination number

10.1142/s1793830920500524 ◽

2020 ◽

Vol 12 (04) ◽

pp. 2050052 ◽

Cited By ~ 1

Author(s):

Lidan Pei ◽

Xiangfeng Pan

Keyword(s):

Positive Integer ◽

Connected Graph ◽

Dominating Set ◽

Domination Number ◽

Dominating Sets ◽

Minimum Cardinality ◽

Number Formula ◽

Simple Connected Graph ◽

Eccentric Distance ◽

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple connected graph. The eccentric distance sum of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the maximum distance from [Formula: see text] to any other vertex and [Formula: see text] is the sum of all distances from [Formula: see text]. A set [Formula: see text] is a distance [Formula: see text]-dominating set of [Formula: see text] if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text]. The minimum cardinality among all distance [Formula: see text]-dominating sets of [Formula: see text] is called the distance [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, the trees among all [Formula: see text]-vertex trees with distance [Formula: see text]-domination number [Formula: see text] having the minimal eccentric distance sum are determined.

Analysis on the Component Connectivity of Enhanced Hypercubes

The Computer Journal ◽

10.1093/comjnl/bxaa122 ◽

2020 ◽

Author(s):

Liqiong Xu ◽

Litao Guo

Keyword(s):

Connected Graph ◽

Interconnection Networks ◽

Unsolved Problem ◽

Reliability Evaluation ◽

Minimum Number ◽

Appl Math

Abstract Reliability evaluation of interconnection networks is of significant importance to the design and maintenance of interconnection networks. The component connectivity is an important parameter for the reliability evaluation of interconnection networks and is a generalization of the traditional connectivity. The $g$-component connectivity $c\kappa _g (G)$ of a non-complete connected graph $G$ is the minimum number of vertices whose deletion results in a graph with at least $g$ components. Determining the $g$-component connectivity is still an unsolved problem in many interconnection networks. Let $Q_{n,k}$ ($1\leq k\leq n-1$) denote the $(n, k)$-enhanced hypercube. In this paper, let $n\geq 7$ and $1\leq k \leq n-5$, we determine $c\kappa _{g}(Q_{n,k}) = g(n + 1) - \frac{1}{2}g(g + 1) + 1$ for $2 \leq g \leq n$. The previous result in Zhao and Yang (2019, Conditional connectivity of folded hypercubes. Discret. Appl. Math., 257, 388–392) is extended.

On the Edge Metric Dimension of Certain Polyphenyl Chains

Journal of Chemistry ◽

10.1155/2021/3855172 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Muhammad Ahsan ◽

Zohaib Zahid ◽

Dalal Alrowaili ◽

Aiyared Iampan ◽

Imran Siddique ◽

...

Keyword(s):

Graph Theory ◽

Connected Graph ◽

Metric Dimension ◽

Chain Graph ◽

Minimum Number ◽

Valence Bonds

The most productive application of graph theory in chemistry is the representation of molecules by the graphs, where vertices and edges of graphs are the atoms and valence bonds between a pair of atoms, respectively. For a vertex w and an edge f = c 1 c 2 of a connected graph G , the minimum number from distances of w with c 1 and c 2 is called the distance between w and f . If for every two distinct edges f 1 , f 2 ∈ E G , there always exists w 1 ∈ W E ⊆ V G such that d f 1 , w 1 ≠ d f 2 , w 1 , then W E is named as an edge metric generator. The minimum number of vertices in W E is known as the edge metric dimension of G . In this paper, we calculate the edge metric dimension of ortho-polyphenyl chain graph O n , meta-polyphenyl chain graph M n , and the linear [n]-tetracene graph T n and also find the edge metric dimension of para-polyphenyl chain graph L n . It has been proved that the edge metric dimension of O n , M n , and T n is bounded, while L n is unbounded.

Upper triangle free detour number of a graph

10.1142/s1793830921500944 ◽

2021 ◽

pp. 2150094

Author(s):

S. Sethu Ramalingam ◽

S. Athisayanathan

Keyword(s):

Free Path ◽

Connected Graph ◽

Proper Subset ◽

Maximum Order ◽

Minimum Order ◽

Geodetic Number ◽

Number Formula ◽

Positive Integers ◽

For any two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the [Formula: see text] path [Formula: see text] is called a [Formula: see text] triangle free path if no three vertices of [Formula: see text] induce a triangle. The triangle free detour distance [Formula: see text] is the length of a longest [Formula: see text] triangle free path in [Formula: see text]. A [Formula: see text] path of length [Formula: see text] is called a [Formula: see text] triangle free detour. A set [Formula: see text] is called a triangle free detour set of [Formula: see text] if every vertex of [Formula: see text] lies on a [Formula: see text] triangle free detour joining a pair of vertices of [Formula: see text]. The triangle free detour number [Formula: see text] of [Formula: see text] is the minimum order of its triangle free detour sets and any triangle free detour set of order [Formula: see text] is a triangle free detour basis of [Formula: see text]. A triangle free detour set [Formula: see text] of [Formula: see text] is called a minimal triangle free detour set if no proper subset of [Formula: see text] is a triangle free detour set of [Formula: see text]. The upper triangle free detour number [Formula: see text] of [Formula: see text] is the maximum order of its minimal triangle free detour sets and any minimal triangle free detour set of order [Formula: see text] is an upper triangle free detour basis of [Formula: see text]. We determine bounds for it and characterize graphs which realize these bounds. For any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. Also, for any four positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], it is shown that there exists a connected graph [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is the upper detour number, [Formula: see text] is the upper detour monophonic number and [Formula: see text] is the upper geodetic number of a graph [Formula: see text].

On the rainbow domination subdivision numbers in graphs

Asian-European Journal of Mathematics ◽

10.1142/s1793557116500182 ◽

2016 ◽

Vol 09 (01) ◽

pp. 1650018 ◽

Cited By ~ 2

Author(s):

N. Dehgardi ◽

M. Falahat ◽

S. M. Sheikholeslami ◽

Abdollah Khodkar

Keyword(s):

Connected Graph ◽

Open Neighborhood ◽

Minimum Weight ◽

Domination Number ◽

Rainbow Domination ◽

Minimum Number ◽

Vertex Set ◽

Number Formula ◽

Domination Subdivision Number ◽

Dominating Function

A [Formula: see text]-rainbow dominating function (2RDF) of a graph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any vertex [Formula: see text] with [Formula: see text] the condition [Formula: see text] is fulfilled, where [Formula: see text] is the open neighborhood of [Formula: see text]. The weight of a 2RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a 2RDF of G. The [Formula: see text]-rainbow domination subdivision number [Formula: see text] is the minimum number of edges that must be subdivided (each edge in [Formula: see text] can be subdivided at most once) in order to increase the 2-rainbow domination number. It is conjectured that for any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. In this paper, we first prove this conjecture for some classes of graphs and then we prove that for any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text].

2-distance vertex-distinguishing total coloring of graphs

10.1142/s1793830918500180 ◽

2018 ◽

Vol 10 (02) ◽

pp. 1850018

Author(s):

Yafang Hu ◽

Weifan Wang

Keyword(s):

Chromatic Number ◽

Maximum Degree ◽

Simple Graph ◽

Total Coloring ◽

Total Chromatic Number ◽

Minimum Number ◽

Number Formula ◽

Proper Total Coloring ◽

A [Formula: see text]-distance vertex-distinguishing total coloring of a graph [Formula: see text] is a proper total coloring of [Formula: see text] such that any pair of vertices at distance [Formula: see text] have distinct sets of colors. The [Formula: see text]-distance vertex-distinguishing total chromatic number [Formula: see text] of [Formula: see text] is the minimum number of colors needed for a [Formula: see text]-distance vertex-distinguishing total coloring of [Formula: see text]. In this paper, we determine the [Formula: see text]-distance vertex-distinguishing total chromatic number of some graphs such as paths, cycles, wheels, trees, unicycle graphs, [Formula: see text], and [Formula: see text]. We conjecture that every simple graph [Formula: see text] with maximum degree [Formula: see text] satisfies [Formula: see text].

ON METRIC DIMENSION OF FUNCTIGRAPHS

10.1142/s1793830912500607 ◽

2013 ◽

Vol 05 (04) ◽

pp. 1250060 ◽

Cited By ~ 2

Author(s):

LINDA EROH ◽

CONG X. KANG ◽

EUNJEONG YI

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Complete Graphs ◽

Minimum Number ◽

Vertex Set

The metric dimension of a graph G, denoted by dim (G), is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let G1and G2be disjoint copies of a graph G and let f : V(G1) → V(G2) be a function. Then a functigraphC(G, f) = (V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = f(u)}. We study how metric dimension behaves in passing from G to C(G, f) by first showing that 2 ≤ dim (C(G, f)) ≤ 2n - 3, if G is a connected graph of order n ≥ 3 and f is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.

Size Ramsey Number of Bounded Degree Graphs for Games

Combinatorics Probability Computing ◽

10.1017/s0963548313000151 ◽

2013 ◽

Vol 22 (4) ◽

pp. 499-516

Author(s):

HEIDI GEBAUER

Keyword(s):

Connected Graph ◽

Maximum Degree ◽

Ramsey Number ◽

Bounded Degree ◽

Minimum Number ◽

Game Theoretic ◽

Bounded Degree Graphs ◽

Constant Maximum

We study Maker/Breaker games on the edges ofsparsegraphs. Maker and Breaker take turns at claiming previously unclaimed edges of a given graphH. Maker aims to occupy a given target graphGand Breaker tries to prevent Maker from achieving his goal. We show that for everydthere is a constantc=c(d)with the property that for every graphGonnvertices of maximum degreedthere is a graphHon at mostcnedges such that Maker has a strategy to occupy a copy ofGin the game onH.This is a result about a game-theoretic variant of the size Ramsey number. For a given graphG,$\hat{r}'(G)$is defined as the smallest numberMfor which there exists a graphHwithMedges such that Maker has a strategy to occupy a copy ofGin the game onH. In this language, our result yields that for every connected graphGof constant maximum degree,$\hat{r}'(G) = \Theta(n)$.Moreover, we can also use our method to settle the corresponding extremal number foruniversalgraphs: for a constantdand for the class${\cal G}_{n}$ofn-vertex graphs of maximum degreed,$s({\cal G}_{n})$denotes the minimum number such that there exists a graphHwithMedges where, foreveryG∈${\cal G}_{n}$, Maker has a strategy to build a copy ofGin the game onH. We obtain that$s({\cal G}_{n}) = \Theta(n^{2 - \frac{2}{d}})$.