scholarly journals The Forcing Restrained Steiner Number of a Graph

2019 ◽  
Vol 8 (6S3) ◽  
pp. 1799-1803
Keyword(s):  
Connected Graph ◽  
Minimum Cardinality ◽  
Unique Minimum ◽  
General Properties

A restrained Steiner set of a connected graph 𝑮 of order 𝒑 ≥ 𝟐 is a set 𝑾 ⊆ 𝑽(𝑮)such that 𝑾 is a Steiner set, and if either 𝑾 = 𝑽 or the subgraph𝑮[𝑽 − 𝑾] inducedby [𝑽 − 𝑾] has no isolated vertices. The restrained Steiner number 𝒔𝒓 𝑮 of 𝑮 isthe minimum cardinality of its restrained Steiner sets and any restrained Steinerset of cardinality 𝒔𝒓 𝑮 is a minimum restrained Steiner set of 𝑮. For a minimum restrained Steiner set 𝑾of 𝑮, a subset 𝑻 ⊆ 𝑾 is called a forcing subset for 𝑾 if 𝑾is the unique minimum restrained Steiner set containing 𝑻. A forcing subset for 𝑾of minimum cardinality is a minimum forcing subset of 𝑾. The forcing restrained Steiner number of 𝑾, denoted by 𝒇𝒓𝒔 𝑾 , is the cardinality of a minimum forcingsubset of 𝑾. The forcing restrained Steiner number of 𝑮, denoted by 𝒇𝒓𝒔 𝑮 is𝒇𝒓𝒔 𝑮 = 𝒎𝒊𝒏⁡{𝒇𝒓𝒔 𝑾 }, where the minimum is taken over all minimum restrainedSteiner sets 𝑾 in 𝑮. Some general properties satisfied by the concept forcing restrained Steiner number are studied. The forcing restrained Steiner number of certain classes of graphs is determined. It is shown that for every pair 𝒂,𝒃 ofintegers with 𝟎 ≤ 𝒂 < 𝒃and 𝒃 ≥ 𝟐, there exists a connected graph 𝑮 such that𝒇𝒓𝒔 𝑮 = 𝒂 and 𝒔𝒓 𝑮 = 𝒃.

THE FORCING EDGE FIXING EDGE-TO-VERTEX MONOPHONIC NUMBER OF A GRAPH

2013 ◽  
Vol 05 (04) ◽  
pp. 1350034
Author(s):  
J. JOHN ◽  
K. UMA SAMUNDESVARI
Keyword(s):  
Connected Graph ◽  
Minimum Cardinality ◽  
General Properties

For a connected graph G = (V, E), a set Se ⊆ E(G)–{e} is called an edge fixing edge-to-vertex monophonic set of an edge e of a connected graph G if every vertex of G lies on an e – f edge-to-vertex monophonic path of G, where f ∈ Se. The edge fixing edge-to-vertex monophonic number mefev(G) of G is the minimum cardinality of its edge fixing edge-to-vertex monophonic sets of an edge e of G. A subset Me ⊆ Se in a connected graph G is called a forcing subset for Se, if Se is the unique edge fixing edge-to-vertex monophonic set of e of G containing Me. A forcing subset for Se of minimum cardinality is a minimum subset of Se. The forcing edge fixing edge-to-vertex monophonic number of G denoted by fefev(G) = min {fefev(Se)}, where the minimum is taken over all cardinality of a minimal edge fixing edge-to-vertex monophonic set of e of G. The forcing edge fixing edge-to-vertex monophonic number of certain classes of graphs is determined and some of its general properties are studied. It is shown that for every integers a and b with 0 ≤ a ≤ b, b ≥ 1, there exists a connected graph G such that fefev(G) = a, mefev(G) = b.

scholarly journals The edge-to-edge geodetic domination number of a graph

2021 ◽  
Vol 40 (3) ◽  
pp. 635-658
Author(s):  
J. John ◽  
V. Sujin Flower
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
General Properties ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Positive Integers

Let G = (V, E) be a connected graph with at least three vertices. A set S ⊆ E(G) is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to-edge geodetic domination number γgee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets. Some general properties satisfied by this concept are studied. Connected graphs of size m with edge-to-edge geodetic domination number 2 or m or m − 1 are characterized. We proved that if G is a connected graph of size m ≥ 4 and Ḡ is also connected, then 4 ≤ γgee(G) + γgee(Ḡ) ≤ 2m − 2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a, b with 2 ≤ a ≤ b, there exists a connected graph G with gee(G) = a and γgee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a ≤ b, there exists a connected graph G with γe(G) = a and γgee(G) = b, where γe(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.

The forcing near geodetic number of a graph

2020 ◽  
Vol 12 (02) ◽  
pp. 2050029
Author(s):  
R. Lenin
Keyword(s):  
Connected Graph ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Unique Minimum ◽  
Geodetic Number ◽  
Forcing Number ◽  
Geodetic Set ◽  
Number Formula ◽  
Positive Integers

A set [Formula: see text] is a near geodetic set if for every [Formula: see text] in [Formula: see text] there exist some [Formula: see text] in [Formula: see text] with [Formula: see text] The near geodetic number [Formula: see text] is the minimum cardinality of a near geodetic set in [Formula: see text] A subset [Formula: see text] of a minimum near geodetic set [Formula: see text] is called the forcing subset of [Formula: see text] if [Formula: see text] is the unique minimum near geodetic set containing [Formula: see text]. The forcing number [Formula: see text] of [Formula: see text] in [Formula: see text] is the minimum cardinality of a forcing subset for [Formula: see text], while the forcing near geodetic number [Formula: see text] of [Formula: see text] is the smallest forcing number among all minimum near geodetic sets of [Formula: see text]. In this paper, we initiate the study of forcing near geodetic number of connected graphs. We characterize graphs with [Formula: see text]. Further, we compare the parameters geodetic number[Formula: see text] near geodetic number[Formula: see text] forcing near geodetic number and we proved that, for every positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a nontrivial connected graph [Formula: see text] with [Formula: see text] [Formula: see text] and [Formula: see text].

scholarly journals The Upper Complement Connected Monophonic Number of a Graph

2019 ◽  
Vol 8 (10S) ◽  
pp. 297-300
Keyword(s):  
Connected Graph ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
General Properties ◽  
Positive Integers

For a connected graph a monophonic set of is said to be a complement connected monophonic set if or the subgraph is connected. The minimum cardinality of a complement connected monophonic set of is the complement connected monophonic number of and is denoted by A complement connected monophonic set in a connected graph is called a minimal complement connected monophonic set if no proper subset of is a complement connected monophonic set of . The upper complement connected monophonic number of is the maximum cardinality of a minimal complement connected monophonic set of . Some general properties under this concept are studied. The upper complement connected monophonic number of some standard graphs are determined. Some of its general properties are studied. It is shown that for any positive integers 2 ≤ a ≤b, there exists a connected graph such that ( ) = a and ( ) =b

The total Steiner number of a graph

2020 ◽  
Vol 12 (03) ◽  
pp. 2050038
Author(s):  
J. John
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
General Properties ◽  
Isolated Vertex

A total Steiner set of [Formula: see text] is a Steiner set [Formula: see text] such that the subgraph [Formula: see text] induced by [Formula: see text] has no isolated vertex. The minimum cardinality of a total Steiner set of [Formula: see text] is the total Steiner number of [Formula: see text] and is denoted by [Formula: see text]. Some general properties satisfied by this concept are studied. Connected graphs of order [Formula: see text] with total Steiner number 2 or 3 are characterized. We partially characterized classes of graphs of order [Formula: see text] with total Steiner number equal to [Formula: see text] or [Formula: see text] or [Formula: see text]. It is shown that [Formula: see text]. It is shown that for every pair k, p of integers with [Formula: see text], there exists a connected graph [Formula: see text] of order [Formula: see text] such that [Formula: see text]. Also, it is shown that for every positive integer [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] of order [Formula: see text] such that [Formula: see text] and [Formula: see text].

scholarly journals The forcing total monophonic number of a graph

2021 ◽  
Vol 40 (2) ◽  
pp. 561-571
Author(s):  
A. P. Santhakumaran ◽  
P. Titus ◽  
K. Ganesamoorthy ◽  
M. Murugan
Keyword(s):  
Connected Graph ◽  
Minimum Cardinality ◽  
Unique Minimum ◽  
Positive Integers

For a connected graph G = (V, E) of order at least two, a subset T of a minimum total monophonic set S of G is a forcing total monophonic subset for S if S is the unique minimum total monophonic set containing T . A forcing total monophonic subset for S of minimum cardinality is a minimum forcing total monophonic subset of S. The forcing total monophonic number ftm(S) in G is the cardinality of a minimum forcing total monophonic subset of S. The forcing total monophonic number of G is ftm(G) = min{ftm(S)}, where the minimum is taken over all minimum total monophonic sets S in G. We determine bounds for it and find the forcing total monophonic number of certain classes of graphs. It is shown that for every pair a, b of positive integers with 0 ≤ a < b and b ≥ a+4, there exists a connected graph G such that ftm(G) = a and mt(G) = b.

scholarly journals Distance Domination and Distance Irredundance in Graphs

10.37236/953 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Adriana Hansberg ◽  
Dirk Meierling ◽  
Lutz Volkmann
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Distance Domination

A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]\neq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$. A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant. The $k$-irredundance number of $G$, denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemańska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.

scholarly journals On the Strong Monophonic Number of a Graph

2019 ◽  
Vol 9 (1) ◽  
pp. 1421-1425
Keyword(s):  
Computational Complexity ◽  
Connected Graph ◽  
General Properties

For a connected graph of order at least two, the strong monophonic problem is to determine a smallest set of vertices of such that, by fixing one monophonic path between each pair of the vertices of all vertices of are covered. In this paper, certain general properties satisfied by the strong monophonic sets are studied. Also, the strong monophonic number of a several families of graphs and computational complexity are determined

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

2020 ◽  
Vol 12 (04) ◽  
pp. 2050052 ◽  
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 ◽  
Distance Formula

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.

scholarly journals The edge geodetic self decomposition number of a graph

2020 ◽  
Author(s):  
John J ◽  
Stalin D
Keyword(s):  
Connected Graph ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Edge Disjoint ◽  
General Properties ◽  
Geodetic Number ◽  
Simple Connected Graph ◽  
Decomposition Number

Let  G = (V, E)  be a simple connected  graph  of order  p and  size q.  A decomposition  of a graph  G is a collection  π  of edge-disjoint sub graphs  G1, G2, ..., Gn  of G such  that every  edge of G belongs to exactly  one Gi , (1 ≤ i ≤ n) . The decomposition  π = {G1, G2, ....Gn } of a connected  graph  G is said to be an edge geodetic self decomposi- tion  if ge (Gi ) = ge (G), (1 ≤ i ≤ n).The maximum  cardinality of π is called the edge geodetic self decomposition  number of G and is denoted by πsge (G), where ge (G) is the edge geodetic number  of G.  Some general properties   satisfied  by  this  concept  are  studied.    Connected  graphs which are edge geodetic self decomposable  are characterized.

Sign in / Sign up

Export Citation Format

Share Document