scholarly journals Connected Weak Edge Detour Number of a Graph

2017 ◽  
Vol 15 (3) ◽  
pp. 43-53
Author(s):  
J M Prabakar ◽  
S Athisayanathan
Keyword(s):  
Connected Graph ◽  
General Properties ◽  
Positive Integers ◽  
Weak Edge

Certain general properties of the detour distance, weak edge detour set, connected weak edge detour set, connected weak edge detour number and connected weak edge detour basis of graphs are studied in this paper. Their relationship with the detour diameter is discussed. It is proved that for each pair of integers k and n with 2 <= k <= n, there exists a connected graph G of order n with cdnw(G)=k. It is also proved that for any three positive integers R,D,k such that k >= D and R < D <= 2R, there exists a connected graph G with radD (G) = R, diamD G = D and cdnw(G)=k.

scholarly journals On the upper geodetic global domination number of a graph

2020 ◽  
Vol 39 (6) ◽  
pp. 1627-1647
Author(s):  
X. Lenin Xaviour ◽  
S. Robinson Chellathurai
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
General Properties ◽  
Geodetic Set ◽  
Positive Integers ◽  
Global Domination Number

A set S of vertices in a connected graph G = (V, E) is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbor in D. A set D is called a global dominating set in G if S is a dominating set of both G and Ḡ. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. A geodetic global dominating set S in G is called a minimal geodetic global dominating set if no proper subset of S is itself a geodetic global dominating set in G. The maximum cardinality of a minimal geodetic global dominating set in G is the upper geodetic global domination number Ῡg+(G) of G. In this paper, the upper geodetic global domination number of certain connected graphs are determined and some of the general properties are studied. It is proved that for all positive integers a, b, p where 3 ≤ a ≤ b < p, there exists a connected graph G such that Ῡg(G) = a, Ῡg+(G) = b and |V (G)| = p.

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.

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

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

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

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.

Edge geodetic self-decomposition in graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050064
Author(s):  
J. John ◽  
D. Stalin
Keyword(s):  
Connected Graph ◽  
Edge Disjoint ◽  
General Properties ◽  
Decomposition Formula ◽  
Simple Connected Graph

Let [Formula: see text] be a simple connected graph of order [Formula: see text] and size [Formula: see text]. A decomposition of a graph [Formula: see text] is a collection of edge-disjoint subgraphs [Formula: see text] of [Formula: see text] such that every edge of [Formula: see text] belongs to exactly one [Formula: see text]. The decomposition [Formula: see text] of a connected graph [Formula: see text] is said to be an edge geodetic self-decomposition if [Formula: see text] for all [Formula: see text]. Some general properties satisfied by this concept are studied.

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.

The open detour number of a graph

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 ◽  
Positive Integers

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].

Distance Vertex Identification in Graphs

2021 ◽  
pp. 2150005
Author(s):  
Gary Chartrand ◽  
Yuya Kono ◽  
Ping Zhang
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Identification Number ◽  
Minimum Number ◽  
Positive Integers ◽  
Distance Formula

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.

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