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 On generalized 3-connectivity of the strong product of graphs

2018 ◽  
Vol 12 (2) ◽  
pp. 297-317
Author(s):  
Encarnación Abajo ◽  
Rocío Casablanca ◽  
Ana Diánez ◽  
Pedro García-Vázquez
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Sharp Bound ◽  
Connected Graphs ◽  
Strong Product ◽  
Generalized Connectivity ◽  
Internally Disjoint Trees ◽  
Product Of Graphs

Let G be a connected graph with n vertices and let k be an integer such that 2 ? k ? n. The generalized connectivity kk(G) of G is the greatest positive integer l for which G contains at least l internally disjoint trees connecting S for any set S ? V (G) of k vertices. We focus on the generalized connectivity of the strong product G1 _ G2 of connected graphs G1 and G2 with at least three vertices and girth at least five, and we prove the sharp bound k3(G1 _ G2) ? k3(G1)_3(G2) + k3(G1) + k3(G2)-1.

Some results about component factors in graphs

2019 ◽  
Vol 53 (3) ◽  
pp. 723-730 ◽  
Author(s):  
Sizhong Zhou
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Connected Graphs ◽  
Spanning Subgraph ◽  
Component Factor ◽  
Path Factor

For a set ℋ of connected graphs, a spanning subgraph H of a graph G is called an ℋ-factor of G if every component of H is isomorphic to a member ofℋ. An H-factor is also referred as a component factor. If each component of H is a star (resp. path), H is called a star (resp. path) factor. By a P≥ k-factor (k positive integer) we mean a path factor in which each component path has at least k vertices (i.e. it has length at least k − 1). A graph G is called a P≥ k-factor covered graph, if for each edge e of G, there is a P≥ k-factor covering e. In this paper, we prove that (1) a graph G has a {K1,1,K1,2, … ,K1,k}-factor if and only if bind(G) ≥ 1/k, where k ≥ 2 is an integer; (2) a connected graph G is a P≥ 2-factor covered graph if bind(G) > 2/3; (3) a connected graph G is a P≥ 3-factor covered graph if bind(G) ≥ 3/2. Furthermore, it is shown that the results in this paper are best possible in some sense.

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.

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].

Herscovici’s conjecture on C2n × G

2020 ◽  
Vol 12 (05) ◽  
pp. 2050071
Author(s):  
A. Lourdusamy ◽  
T. Mathivanan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Adjacent Vertex ◽  
Connected Graphs ◽  
Number Formula ◽  
Pebbling Number

The [Formula: see text]-pebbling number, [Formula: see text], of a connected graph [Formula: see text], is the smallest positive integer such that from every placement of [Formula: see text] pebbles, [Formula: see text] pebbles can be moved to any specified target vertex by a sequence of pebbling moves, each move taking two pebbles off a vertex and placing one on an adjacent vertex. A graph [Formula: see text] satisfies the [Formula: see text]-pebbling property if [Formula: see text] pebbles can be moved to any specified vertex when the total starting number of pebbles is [Formula: see text], where [Formula: see text] is the number of vertices with at least one pebble. We show that the cycle [Formula: see text] satisfies the [Formula: see text]-pebbling property. Herscovici conjectured that for any connected graphs [Formula: see text] and [Formula: see text], [Formula: see text]. We prove Herscovici’s conjecture is true, when [Formula: see text] is an even cycle and for variety of graphs [Formula: see text] which satisfy the [Formula: see text]-pebbling property.

ON EDGE-CONNECTIVITY AND SUPER EDGE-CONNECTIVITY OF INTERCONNECTION NETWORKS MODELED BY PRODUCT GRAPHS

2010 ◽  
Vol 02 (02) ◽  
pp. 143-150
Author(s):  
CHUNXIANG WANG
Keyword(s):  
Connected Graph ◽  
Interconnection Networks ◽  
Minimum Degree ◽  
Edge Connectivity ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Product Graph ◽  
Product Graphs

The super edge-connectivity λ′ of a connected graph G is the minimum cardinality of an edge-cut F in G such that every component of G–F contains at least two vertices. Let two connected graphs Gm and Gp have m and p vertices, minimum degree δ(Gm) and δ(Gp), edge-connectivity λ(Gm) and λ(Gp), respectively. This paper shows that min {pλ(Gm), λ(Gp) + δ(Gm), δ(Gm)(λ(Gp) + 1), (δ(Gm) + 1)λ(Gp)} ≤ λ(Gm * Gp) ≤ δ(Gm) + δ(Gp), where the product graph Gm * Gp of two given graphs Gm and Gp, defined by J. C. Bermond et al. [J. Combin. Theory B36 (1984) 32–48] in the context of the so-called (△, D)-problem, is one interesting model in the design of large reliable networks. Moreover, this paper determines λ′(Gm * Gp) ≤ min {pδ(Gm), ξ(Gp) + 2δ(Gm)} and λ′(G1 ⊕ G2) ≥ min {n, λ1 + λ2} if δ1 = δ2.

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.

Wiener Index of k-Connected Graphs

2021 ◽  
pp. 2142005
Author(s):  
Xiang Qin ◽  
Yanhua Zhao ◽  
Baoyindureng Wu
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Wiener Index ◽  
Index Formula ◽  
Connected Graphs ◽  
Sum Of Distances

The Wiener index [Formula: see text] of a connected graph [Formula: see text] is the sum of distances of all pairs of vertices in [Formula: see text]. In this paper, we show that for any even positive integer [Formula: see text], and [Formula: see text], if [Formula: see text] is a [Formula: see text]-connected graph of order [Formula: see text], then [Formula: see text], where [Formula: see text] is the [Formula: see text]th power of a graph [Formula: see text]. This partially answers an old problem of Gutman and Zhang.

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