scholarly journals Removable Cycles Avoiding Two Connected Subgraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Y. M. Borse ◽  
B. N. Waphare
Keyword(s):  
Connected Graph ◽  
Sufficient Condition ◽  
Edge Disjoint ◽  
Connected Subgraphs

We provide a sufficient condition for the existence of a cycle in a connected graph which is edge-disjoint from two connected subgraphs and of such that is connected.

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.

scholarly journals Detection number of bipartite graphs and cubic graphs

2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
International Audience ◽  
Np Complete

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.

Peg solitaire on graphs with large maximum degree

2021 ◽  
pp. 2250057
Author(s):  
Naoki Matsumoto
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Terminal State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Combinatorial Game ◽  
Starting State ◽  
Large Maximum ◽  
Necessary And Sufficient

In 2011, Beeler and Hoilman introduced the peg solitaire on graphs. The peg solitaire on a connected graph is a one-player combinatorial game starting with exactly one hole in a vertex and pegs in all other vertices and removing all pegs but exactly one by a sequence of jumps; for a path [Formula: see text], if there are pegs in [Formula: see text] and [Formula: see text] and exists a hole in [Formula: see text], then [Formula: see text] can jump over [Formula: see text] into [Formula: see text], and after that, the peg in [Formula: see text] is removed. A problem of interest in the game is to characterize solvable (respectively, freely solvable) graphs, where a graph is solvable (respectively, freely solvable) if for some (respectively, any) vertex [Formula: see text], starting with a hole [Formula: see text], a terminal state consisting of a single peg can be obtained from the starting state by a sequence of jumps. In this paper, we consider the peg solitaire on graphs with large maximum degree. In particular, we show the necessary and sufficient condition for a graph with large maximum degree to be solvable in terms of the number of pendant vertices adjacent to a vertex of maximum degree. It is a notable point that this paper deals with a question of Beeler and Walvoort whether a non-solvable condition of trees can be extended to other graphs.

An Optimal Disjoint Pair of Additive Spanners for the 3D Grid

1998 ◽  
Vol 08 (02) ◽  
pp. 251-258
Author(s):  
David W. Krumme
Keyword(s):  
Connected Graph ◽  
Three Dimensional ◽  
Connected Subgraph ◽  
Interesting Problem ◽  
Constant Delay ◽  
Edge Disjoint ◽  
Disjoint Pair ◽  
Parallel Network ◽  
And Function

A spanner of a connected graph G is a spanning connected subgraph S. If DG(u, v) and DS(u, v) denote the distance between vertices u and v in G and S, respectively, then S is an f(x)-spanner if and only if DS(u, v) ≤ f(DG(u, v)). The value f(x) - x is the delay of the spanner. An additive spanner is one with constant delay. Given a graph G and function f(x), an interesting problem is to partition the edges of G so as to define edge-disjoint f(x)-spanners of G. This paper exhibits a pair of (x + 4)-spanners for the infinite three dimensional grid, and shows that 4 is the least integer K for which there exists an edge-disjoint pair of delay-K spanners. spanner, grid, additive spanner, parallel network.

scholarly journals A Similarity Measurement Method Based on Graph Kernel for Disconnected Graphs

2019 ◽  
Author(s):  
Jun Gao ◽  
Jianliang Gao
Keyword(s):  
Real World ◽  
Connected Graph ◽  
Measurement Method ◽  
Dimensional Space ◽  
Similarity Measurement ◽  
Graph Kernel ◽  
The Real ◽  
Connected Subgraphs ◽  
Low Dimensional ◽  
Disconnected Graphs

Disconnected graphs are very common in the real world. However, most existing methods for graph similarity focus on connected graph. In this paper, we propose an effective approach for measuring the similarity of disconnected graphs. By embedding connected subgraphs with graph kernel, we obtain the feature vectors in low dimensional space. Then, we match the subgraphs and weigh the similarity of matched subgraphs. Finally, an intuitive example shows the feasibility of the method.

scholarly journals A study on the pendant number of graph products

2019 ◽  
Vol 11 (1) ◽  
pp. 24-40
Author(s):  
Jomon K. Sebastian ◽  
Joseph Varghese Kureethara ◽  
Sudev Naduvath ◽  
Charles Dominic
Keyword(s):  
Natural Number ◽  
Connected Graph ◽  
Disjoint Paths ◽  
Graph Products ◽  
Path Decomposition ◽  
Edge Disjoint ◽  
Minimum Number ◽  
Number Π

Abstract A path decomposition of a graph is a collection of its edge disjoint paths whose union is G. The pendant number Πp is the minimum number of end vertices of paths in a path decomposition of G. In this paper, we determine the pendant number of corona products and rooted products of paths and cycles and obtain some bounds for the pendant number for some specific derived graphs. Further, for any natural number n, the existence of a connected graph with pendant number n has also been established.

scholarly journals Power of 2 Decomposition of a Complete Tripartite Graph K2,4,M and a Special Butterfly Graph

2020 ◽  
Vol 9 (3) ◽  
pp. 3973-3976
Keyword(s):  
Simple Graph ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Edge Disjoint ◽  
Tripartite Graph ◽  
Necessary And Sufficient ◽  
Complete Tripartite Graph

Let G be a finite, connected simple graph with p vertices and q edges. If G1 , G2 ,…, Gn are connected edge-disjoint subgraphs of G with E(G) = E(G1 )  E(G2 )  …  E(Gn) , then {G1 , G2 , …, Gn} is said to be a decomposition of G. A graph G is said to have Power of 2 Decomposition if G can be decomposed into edge-disjoint subgraphs G G G n  2 4 2 , ,..., such that each G i 2 is connected and ( ) 2 , i E Gi  for 1  i  n. Clearly, 2[2 1] n q . In this paper, we investigate the necessary and sufficient condition for a complete tripartite graph K2,4,m and a Special Butterfly graph           3 2 5 BF 2m 1 to accept Power of 2 Decomposition.

Some results for the two disjoint connected dominating sets problem

2019 ◽  
Vol 11 (06) ◽  
pp. 1950065
Author(s):  
Xianliang Liu ◽  
Zishen Yang ◽  
Wei Wang
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Sufficient Condition ◽  
Dominating Sets ◽  
Cubic Graph ◽  
Connected Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Dominating Set Problem

As a variant of minimum connected dominating set problem, two disjoint connected dominating sets (DCDS) problem is to ask whether there are two DCDS [Formula: see text] in a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], and if not, how to add an edge subset with minimum cardinality such that the new graph has a pair of DCDS. The two DCDS problem is so hard that it is NP-hard on trees. In this paper, if the vertex set [Formula: see text] of a connected graph [Formula: see text] can be partitioned into two DCDS of [Formula: see text], then it is called a DCDS graph. First, a necessary but not sufficient condition is proposed for cubic (3-regular) graph to be a DCDS graph. To be exact, if a cubic graph is a DCDS graph, there are at most four disjoint triangles in it. Next, if a connected graph [Formula: see text] is a DCDS graph, a simple but nontrivial upper bound [Formula: see text] of the girth [Formula: see text] is presented.

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