Some Domination Parameters of Halin Graph with Perfect K-ary Tree and its Application

2020 ◽  
Vol 14 (6) ◽  
Keyword(s):  
Halin Graph

scholarly journals Hamiltonicity of Cubic 3-Connected k-Halin Graphs

10.37236/3188 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Shabnam Malik ◽  
Ahmad Mahmood Qureshi ◽  
Tudor Zamfirescu
Keyword(s):  
Connected Graphs ◽  
Halin Graph ◽  
Halin Graphs

We investigate here how far we can extend the notion of a Halin graph such that hamiltonicity is preserved. Let $H = T \cup C$ be a Halin graph, $T$ being a tree and $C$ the outer cycle. A $k$-Halin graph $G$ can be obtained from $H$ by adding edges while keeping planarity, joining vertices of $H - C$, such that $G - C$ has at most $k$ cycles. We prove that, in the class of cubic $3$-connected graphs, all $14$-Halin graphs are hamiltonian and all $7$-Halin graphs are $1$-edge hamiltonian. These results are best possible.

scholarly journals Acyclic chromatic index of fully subdivided graphs and Halin graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Manu Basavaraju
Keyword(s):  
Rooted Tree ◽  
Edge Coloring ◽  
Direct Proof ◽  
Chromatic Index ◽  
Acyclic Edge Coloring ◽  
Halin Graph ◽  
Minimum Number ◽  
Vertex Set ◽  
International Audience ◽  
Acyclic Chromatic Index

Graph Theory International audience An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). A graph G is called fully subdivided if it is obtained from another graph H by replacing every edge by a path of length at least two. Fully subdivided graphs are known to be acyclically edge colorable using Δ+1 colors since they are properly contained in 2-degenerate graphs which are acyclically edge colorable using Δ+1 colors. Muthu, Narayanan and Subramanian gave a simple direct proof of this fact for the fully subdivided graphs. Fiamcik has shown that if we subdivide every edge in a cubic graph with at most two exceptions to get a graph G, then a'(G)=3. In this paper we generalise the bound to Δ for all fully subdivided graphs improving the result of Muthu et al. In particular, we prove that if G is a fully subdivided graph and Δ(G) ≥3, then a'(G)=Δ(G). Consider a graph G=(V,E), with E=E(T) ∪E(C) where T is a rooted tree on the vertex set V and C is a simple cycle on the leaves of T. Such a graph G is called a Halin graph if G has a planar embedding and T has no vertices of degree 2. Let Kn denote a complete graph on n vertices. Let G be a Halin graph with maximum degree Δ. We prove that, a'(G) = 5 if G is K4, 4 if Δ = 3 and G is not K4, and Δ otherwise.

Planning rendezvous using the Halin graph in wireless sensor networks

2012 ◽  
Vol 2 (3) ◽  
pp. 222 ◽  
Author(s):  
R.-S. Chang ◽  
S.-H. Wang ◽  
S.-L. Tsai ◽  
W.-P. Yang
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Wireless Sensor ◽  
Halin Graph

scholarly journals Listing all spanning trees in Halin graphs — sequential and Parallel view

2018 ◽  
Vol 10 (01) ◽  
pp. 1850005
Author(s):  
K. Krishna Mohan Reddy ◽  
P. Renjith ◽  
N. Sadagopan
Keyword(s):  
Parallel Algorithm ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Combinatorial Problem ◽  
Halin Graph ◽  
Labeled Graph ◽  
Number Of Leaves ◽  
Acyclic Subgraph ◽  
Number Of Spanning Trees ◽  
Halin Graphs

For a connected labeled graph [Formula: see text], a spanning tree [Formula: see text] is a connected and acyclic subgraph that spans all vertices of [Formula: see text]. In this paper, we consider a classical combinatorial problem which is to list all spanning trees of [Formula: see text]. A Halin graph is a graph obtained from a tree with no degree two vertices and by joining all leaves with a cycle. We present a sequential and parallel algorithm to enumerate all spanning trees in Halin graphs. Our approach enumerates without repetitions and we make use of [Formula: see text] processors for parallel algorithmics, where [Formula: see text] and [Formula: see text] are the depth, the number of leaves, respectively, of the Halin graph. We also prove that the number of spanning trees in Halin graphs is [Formula: see text].

Hamiltonian Properties of Generalized Halin Graphs

2009 ◽  
Vol 52 (3) ◽  
pp. 416-423 ◽  
Author(s):  
Shabnam Malik ◽  
Ahmad Mahmood Qureshi ◽  
Tudor Zamfirescu
Keyword(s):  
Cyclic Order ◽  
Halin Graph ◽  
Hamiltonian Properties ◽  
Halin Graphs

AbstractA Halin graph is a graph H = T ∪ C, where T is a tree with no vertex of degree two, and C is a cycle connecting the end-vertices of T in the cyclic order determined by a plane embedding of T. In this paper, we define classes of generalized Halin graphs, called k-Halin graphs, and investigate their Hamiltonian properties.

On independent domination parameters of some special families of Halin graph

2019 ◽  
Vol 22 (6) ◽  
pp. 1113-1119
Author(s):  
D. Anandhababu ◽  
M. Priyadharshini ◽  
N. Parvathi
Keyword(s):  
Halin Graph ◽  
Independent Domination

An O(n) Time Algorithm to Find Cubic Subgraph in a Halin Graph

2013 ◽  
Vol 380-384 ◽  
pp. 1318-1322
Author(s):  
Ding Jun Lou ◽  
Jun Fu Liu
Keyword(s):  
Time Algorithm ◽  
Halin Graph ◽  
Complete Problem ◽  
Np Complete ◽  
Regular Subgraph ◽  
General Graphs

The 3-Regular Subgraph Problem is: Given a graph G = (V, E), can we find a subgraph H = (V, E) in G such that for each vertex u in V, , where is the degree of u in H? This problem is an NP-complete problem for general graphs. In this paper, we design an O(n) time algorithm to solve The 3-Regular Subgraph Problem for a Halin graph H, where n is the number of vertices of H. Given a Halin graph H, if there is a cubic subgraph G in H, then our algorithm will find G and give an answer Yes, otherwise our algorithm will give an answer No. We also prove the correctness of this algorithm.

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