Maximum incomplete recursive circulants in graph embeddings

2015 ◽  
Vol 07 (04) ◽  
pp. 1550053
Author(s):  
R. Sundara Rajan ◽  
Indra Rajasingh ◽  
Paul Manuel ◽  
Mirka Miller ◽  
T. M. Rajalaxmi
Keyword(s):  
Parallel Algorithms ◽  
Small Diameter ◽  
Performance Degradation ◽  
Binary Trees ◽  
Graph Embeddings ◽  
Binomial Trees ◽  
Np Complete ◽  
Good Support ◽  
Special Classes

An incomplete recursive circulant possesses virtually every advantage of a complete recursive circulant, including simple deadlock-free routing, a small diameter, a good support of parallel algorithms, and so on. It is natural to reconfigure a faulty recursive circulant into a maximum incomplete recursive circulant so as to lower potential performance degradation. For [Formula: see text], the maximum incomplete subgraph problem is to identify a subgraph [Formula: see text] of a graph [Formula: see text] on [Formula: see text] vertices having the maximum number of edges among all subgraphs on [Formula: see text] vertices and is NP-complete. In this paper we identify maximum incomplete recursive circulants and use them as a tool to compute the exact wirelength of embedding recursive circulants into special classes of trees, such as [Formula: see text]-rooted complete binary trees, [Formula: see text]-rooted sibling trees, binomial trees, certain caterpillars and path.

scholarly journals A note on parallel algorithms for optimal h-v drawings of binary trees

1998 ◽  
Vol 9 (3) ◽  
pp. 145-158 ◽  
Author(s):  
Panagiotis T. Metaxas ◽  
Grammati E. Pantziou ◽  
Antonis Symvonis
Keyword(s):  
Parallel Algorithms ◽  
Binary Trees

A UNIFIED APPROACH TO PARALLEL ALGORITHMS FOR THE DOMATIC PARTITION PROBLEM ON SPECIAL CLASSES OF PERFECT GRAPHS

1993 ◽  
Vol 03 (03) ◽  
pp. 233-241
Author(s):  
A. RAMAN ◽  
C. PANDU RANGAN
Keyword(s):  
Parallel Algorithms ◽  
Dominating Set ◽  
Interval Graphs ◽  
Perfect Graphs ◽  
Partition Problem ◽  
Dominating Sets ◽  
Unified Approach ◽  
Block Graphs ◽  
Vertex Set ◽  
Special Classes

A set of vertices D is a dominating set of a graph G=(V, E) if every vertex in V−D is adjacent to at least one vertex in D. The domatic partition of G is the partition of the vertex set V into a maximum number of dominating sets. In this paper, we present efficient parallel algorithms for finding the domatic partition of Interval graphs, Block graphs and K-trees.

scholarly journals Strong edge geodetic problem in networks

2017 ◽  
Vol 15 (1) ◽  
pp. 1225-1235 ◽  
Author(s):  
Paul Manuel ◽  
Sandi Klavžar ◽  
Antony Xavier ◽  
Andrew Arokiaraj ◽  
Elizabeth Thomas
Keyword(s):  
Graph Theory ◽  
Exact Solutions ◽  
Upper Bounds ◽  
Binary Trees ◽  
Lower And Upper Bounds ◽  
Transport Networks ◽  
Covering Problems ◽  
Geodetic Number ◽  
Block Graphs ◽  
Np Complete

Abstract Geodesic covering problems form a widely researched topic in graph theory. One such problem is geodetic problem introduced by Harary et al. [Math. Comput. Modelling, 1993, 17, 89-95]. Here we introduce a variation of the geodetic problem and call it strong edge geodetic problem. We illustrate how this problem is evolved from social transport networks. It is shown that the strong edge geodetic problem is NP-complete. We derive lower and upper bounds for the strong edge geodetic number and demonstrate that these bounds are sharp. We produce exact solutions for trees, block graphs, silicate networks and glued binary trees without randomization.

scholarly journals On the Hosoya Indices of Bicyclic Graphs with Small Diameter

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Tingzeng Wu ◽  
Yong Yu
Keyword(s):  
Small Diameter ◽  
Sharp Bound ◽  
Hosoya Index ◽  
Np Complete ◽  
Bicyclic Graphs

Let G be a graph. The Hosoya index of G , denoted by z G , is defined as the total number of its matchings. The computation of z G is NP-Complete. Wagner and Gutman pointed out that it is difficult to obtain results of the maximum Hosoya index among tree-like graphs with given diameter. In this paper, we focus on the problem, and a sharp bound of Hosoya indices of all bicyclic graphs with diameter of 3 is determined.

scholarly journals Coloring Problems on Bipartite Graphs of Small Diameter

10.37236/9931 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Victor A. Campos ◽  
Guilherme C.M. Gomes ◽  
Allen Ibiapina ◽  
Raul Lopes ◽  
Ignasi Sau ◽  
...  
Keyword(s):  
Small Diameter ◽  
Bipartite Graphs ◽  
List Coloring ◽  
Coloring Problem ◽  
Np Completeness ◽  
Np Complete

We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we investigate the $k$-List Coloring, $k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with diameter at most $d$, proving $\textsf{NP}$-completeness in most cases, and leaving open only the List $3$-Coloring and $3$-Precoloring Extension problems when $d=3$. Some of these results are obtained $\textsc{through}$ a proof that the Surjective $C_6$-Homomorphism problem is $\textsf{NP}$-complete on bipartite graphs with diameter at most four. Although the latter result has been already proved [Vikas, 2017], we present ours as an alternative simpler one. As a byproduct, we also get that $3$-Biclique Partition is $\textsf{NP}$-complete. An attempt to prove this result was presented in [Fleischner, Mujuni, Paulusma, and Szeider, 2009], but there was a flaw in their proof, which we identify and discuss here. Finally, we prove that the $3$-Fall Coloring problem is $\textsf{NP}$-complete on bipartite graphs with diameter at most four, and prove that $\textsf{NP}$-completeness for diameter three would also imply $\textsf{NP}$-completeness of $3$-Precoloring Extension on diameter three, thus closing the previously mentioned open cases. This would also answer a question posed in [Kratochvíl, Tuza, and Voigt, 2002].

scholarly journals Complexity aspects of the computation of the rank of a graph

2014 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
Igor Ramos ◽  
Vinícius F. Santos ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Convex Hull ◽  
Upper Bound ◽  
Convex Set ◽  
Small Diameter ◽  
Independent Set ◽  
Special Issue ◽  
Undirected Graphs ◽  
Radon Number ◽  
International Audience ◽  
Np Complete

Special issue PRIMA 2013 International audience We consider the P₃-convexity on simple undirected graphs, in which a set of vertices S is convex if no vertex outside S has two or more neighbors in S. The convex hull H(S) of a set S is the smallest convex set containing S as a subset. A set S is a convexly independent set if v \not ∈ H(S\setminus \v\) for all v in S. The rank \rk(G) of a graph is the size of the largest convexly independent set. In this paper we consider the complexity of determining \rk(G). We show that the problem is NP-complete even for split or bipartite graphs with small diameter. We also show how to determine \rk(G) in polynomial time for the well structured classes of graphs of trees and threshold graphs. Finally, we give a tight upper bound for \rk(G), which in turn gives a tight upper bound for the Radon number as byproduct, which is the same obtained before by Henning, Rautenbach and Schäfer. Additionally, we briefly show that the problem is NP-complete also in the monophonic convexity.

EFFICIENT MULTIPLE-ITEM BROADCAST IN THE LogP MODEL

1993 ◽  
Vol 03 (04) ◽  
pp. 407-417 ◽  
Author(s):  
RAMESH SUBRAMONIAN ◽  
NARAYAN VENKATASUBRAMANYAN
Keyword(s):  
Parallel Algorithms ◽  
Optimal Solution ◽  
Performance Degradation ◽  
Minimum Time ◽  
Message Transmission ◽  
Multiple Item ◽  
Logp Model ◽  
Point To Point ◽  
Dedicated Hardware

A common arising problem in many parallel algorithms is broadcast. Many multiprocessors do not have dedicated hardware for this purpose. In this paper, we address the problem of simulating multiple-item broadcast by point-to-point message transmission, where a source processor has many messages which it wishes to disseminate to P−1 other processors. In a step, a processor can send at most one item from among those in its possession and receive at most one item. An item is received at most L steps after it is sent. The goal is to find a schedule that achieves the broadcast in minimum time. We improve on previous results by developing a simpler and almost optimal solution which makes no assumptions about L or P. We also provide a bound on the performance degradation when the latency, L, is allowed to vary.

Sign in / Sign up

Export Citation Format

Share Document