The forcing total monophonic number of a graph

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.

The forcing near geodetic number of a graph

10.1142/s1793830920500299 ◽

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 ◽

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].

The open detour number of a graph

10.1142/s1793830920500883 ◽

2020 ◽

pp. 2050088

Author(s):

J. John ◽

V. R. Sunil Kumar

Keyword(s):

Connected Graph ◽

Internal Vertex ◽

Minimum Cardinality ◽

Connected Graphs ◽

Geodetic Number ◽

Number Formula ◽

A set [Formula: see text] is called an open detour set of [Formula: see text] if for each vertex [Formula: see text] in [Formula: see text], either (1) [Formula: see text] is a detour simplicial vertex of [Formula: see text] and [Formula: see text] or (2) [Formula: see text] is an internal vertex of an [Formula: see text]-[Formula: see text] detour for some [Formula: see text]. An open detour set of minimum cardinality is called a minimum open detour set and this cardinality is the open detour number of [Formula: see text], denoted by [Formula: see text]. Connected graphs of order [Formula: see text] with open detour number [Formula: see text] or [Formula: see text] are characterized. It is shown that for any two positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the detour number of [Formula: see text]. It is also shown that for every pair of positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the open geodetic number of [Formula: see text].

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

Proyecciones (Antofagasta) ◽

10.22199/issn.0717-6279-4057 ◽

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 ◽

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 upper connected vertex detour number of a graph

Filomat ◽

10.2298/fil1202379s ◽

2012 ◽

Vol 26 (2) ◽

pp. 379-388 ◽

Author(s):

A.P. Santhakumaran ◽

P. Titus

Keyword(s):

Connected Graph ◽

Proper Subset ◽

Maximum Cardinality ◽

Minimum Cardinality ◽

Positive Integers ◽

Connected Vertex ◽

Special Classes

For vertices x and y in a connected graph G = (V, E) of order at least two, the detour distance D(x, y) is the length of the longest x ? y path in G: An x ? y path of length D(x, y) is called an x ? y detour. For any vertex x in G, a set S ? V is an x-detour set of G if each vertex v ? V lies on an x ? y detour for some element y in S: The minimum cardinality of an x-detour set of G is defined as the x-detour number of G; denoted by dx(G): An x-detour set of cardinality dx(G) is called a dx-set of G: A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cdx(G). A connected x-detour set of cardinality cdx(G) is called a cdx-set of G. A connected x-detour set Sx is called a minimal connected x-detour set if no proper subset of Sx is a connected x-detour set. The upper connected x-detour number, denoted by cd+ x (G), is defined as the maximum cardinality of a minimal connected x-detour set of G: We determine bounds for cd+ x (G) and find the same for some special classes of graphs. For any three integers a; b and c with 2 ? a < b ? c, there is a connected graph G with dx(G) = a; cdx(G) = b and cd+ x (G) = c for some vertex x in G: It is shown that for positive integers R,D and n ? 3 with R < D ? 2R; there exists a connected graph G with detour radius R; detour diameter D and cd+ x (G) = n for some vertex x in G.

International Journal of Innovative Technology and Exploring Engineering - Special Issue ◽

The Upper Complement Connected Monophonic Number of a Graph

10.35940/ijitee.j1053.08810s19 ◽

2019 ◽

Vol 8 (10S) ◽

pp. 297-300

Keyword(s):

Connected Graph ◽

Proper Subset ◽

Maximum Cardinality ◽

Minimum Cardinality ◽

General Properties ◽

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

EXTREME STEINER GRAPHS

10.1142/s1793830912500292 ◽

2012 ◽

Vol 04 (02) ◽

pp. 1250029 ◽

Author(s):

A. P. SANTHAKUMARAN

Keyword(s):

Connected Graph ◽

Steiner Tree ◽

Minimum Order ◽

Minimum Cardinality ◽

Geodetic Number ◽

Geodetic Set ◽

For a connected graph G of order p ≥ 2 and a set W ⊆ V(G), a tree T contained in G is a Steiner tree with respect to W if T is a tree of minimum order with W ⊆ V(T). The set S(W) consists of all vertices in G that lie on some Steiner tree with respect to W. The set W is a Steiner set for G if S(W) = V(G). The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. A geodetic set of G is a set S of vertices such that every vertex of G is contained in a geodesic joining some pair of vertices of S. The geodetic number g(G) of G is the minimum cardinality of its geodetic sets and any geodetic set of cardinality g(G) is a minimum geodetic set of G. A vertex v is an extreme vertex of a graph G if the subgraph induced by its neighbors is complete. The number of extreme vertices in G is its extreme order ex (G). A graph G is an extreme Steiner graph if s(G) = ex (G), and an extreme geodesic graph if g(G) = ex (G). Extreme Steiner graphs of order p with Steiner number p - 1 are characterized. It is shown that every pair a, b of integers with 0 ≤ a ≤ b is realizable as the extreme order and Steiner number, respectively, of some graph. For positive integers r, d and l ≥ 2 with r < d ≤ 2r, it is shown that there exists an extreme Steiner graph G of radius r, diameter d, and Steiner number l. For integers p, d and k with 2 ≤ d < p, 2 ≤ k < p and p - d - k + 2 ≥ 0, there exists an extreme Steiner graph G of order p, diameter d and Steiner number k. It is shown that for every pair a, b of integers with 3 ≤ a < b and b = a + 1, there exists an extreme Steiner graph G with s(G) = a and g(G) = b that is not an extreme geodesic graph. It is shown that for every pair a, b of integers with 3 ≤ a < b, there exists an extreme geodesic graph G with g(G) = a and s(G) = b that is not an extreme Steiner graph.

THE LINEAR GEODETIC NUMBER OF A GRAPH

10.1142/s1793830911001279 ◽

2011 ◽

Vol 03 (03) ◽

pp. 357-368 ◽

Author(s):

A. P. SANTHAKUMARAN ◽

T. JEBARAJ ◽

S. V. ULLAS CHANDRAN

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Minimum Cardinality ◽

Connected Graphs ◽

Geodetic Number ◽

Geodetic Set ◽

For a connected graph G of order n, an ordered set S = {u1, u2, …, uk} of vertices in G is a linear geodetic set of G if for each vertex x in G, there exists an index i, 1 ≤ i < k such that x lies on a ui - ui + 1 geodesic on G, and a linear geodetic set of minimum cardinality is the linear geodetic number gl(G). The linear geodetic numbers of certain standard graphs are obtained. It is shown that if G is a graph of order n and diameter d, then gl(G) ≤ n - d + 1 and this bound is sharp. For positive integers r, d and k ≥ 2 with r < d ≤ 2r, there exists a connected graph G with rad G = r, diam G = d and gl(G) = k. Also, for integers n, d and k with 2 ≤ d < n, 2 ≤ k ≤ n - d + 1, there exists a connected graph G of order n, diameter d and gl(G) = k. We characterize connected graphs G of order n with gl(G) = n and gl(G) = n - 1. It is shown that for each pair a, b of integers with 3 ≤ a ≤ b, there is a connected graph G with g(G) = a and gl(G) = b. We also discuss how the linear geodetic number of a graph is affected by adding a pendent edge to the graph.

The Forcing Restrained Steiner Number of a Graph

International Journal of Engineering and Advanced Technology - Regular Issue ◽

10.35940/ijeat.f1382.0986s319 ◽

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.

Extreme Monophonic Graphs and Extreme Geodesic Graphs

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.47.2016.2045 ◽

2016 ◽

Vol 47 (4) ◽

pp. 393-404

Author(s):

P. Titus ◽

A.P Santhakumaran

Keyword(s):

Connected Graph ◽

Minimum Cardinality ◽

Characterization Result ◽

Geodetic Number ◽

Geodetic Set ◽

Positive Integers ◽

Chordless Path

For a connected graph $G=(V,E)$ of order at least two, a chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path $P$ is called a monophonic path if it is a chordless path. A monophonic set of $G$ is a set $S$ of vertices such that every vertex of $G$ lies on a monophonic path joining some pair of vertices in $S$. The monophonic number of $G$ is the minimum cardinality of its monophonic sets and is denoted by $m(G)$. A geodetic set of $G$ is a set $S$ of vertices such that every vertex of $G$ lies on a geodesic joining some pair of vertices in $S$. The geodetic number of $G$ is the minimum cardinality of its geodetic sets and is denoted by $g(G)$. The number of extreme vertices in $G$ is its extreme order $ex(G)$. A graph $G$ is an extreme monophonic graph if $m(G)=ex(G)$ and an extreme geodesic graph if $g(G)=ex(G)$. Extreme monophonic graphs of order $p$ with monophonic number $p$ and $p-1$ are characterized. It is shown that every pair $a,b$ of integers with $0 \leq a \leq b$ is realized as the extreme order and monophonic number, respectively, of some graph. For positive integers $r,d$ and $k \geq 3$ with $r < d$, it is shown that there exists an extreme monophonic graph $G$ of monophonic radius $r$, monophonic diameter $d$, and monophonic number $k$. Also, we give a characterization result for a graph $G$ which is both extreme geodesic and extreme monophonic.

The upper connected edge geodetic number of a graph

Filomat ◽

10.2298/fil1201131s ◽

2012 ◽

Vol 26 (1) ◽

pp. 131-141 ◽

Author(s):

A.P. Santhakumaran ◽

J. John

Keyword(s):

Connected Graph ◽

Proper Subset ◽

Maximum Cardinality ◽

Minimum Order ◽

Minimum Cardinality ◽

Geodetic Number ◽

Geodetic Set ◽

For a non-trivial connected graph G, a set S ? V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected edge geodetic set of G is the connected edge geodetic number of G and is denoted by g1c(G). A connected edge geodetic set of cardinality g1c(G) is called a g1c- set of G or connected edge geodetic basis of G. A connected edge geodetic set S in a connected graph G is called a minimal connected edge geodetic set if no proper subset of S is a connected edge geodetic set of G. The upper connected edge geodetic number g+ 1c(G) is the maximum cardinality of a minimal connected edge geodetic set of G. Graphs G of order p for which g1c(G) = g+1c = p are characterized. For positive integers r,d and n ( d + 1 with r ? d ? 2r, there exists a connected graph of radius r, diameter d and upper connected edge geodetic number n. It is shown for any positive integers 2 ? a < b ? c, there exists a connected graph G such that g1(G) = a; g1c(G) = b and g+ 1c(G) = c.