scholarly journals Characterization of the Congestion Lemma on Layout Computation

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Jia-Bao Liu ◽  
Arul Jeya Shalini ◽  
Micheal Arockiaraj ◽  
J. Nancy Delaila
Keyword(s):  
Graph Embedding ◽  
Complete Graphs ◽  
Bijective Function ◽  
Circulant Networks

An embedding of a guest network G N into a host network H N is to find a suitable bijective function between the vertices of the guest and the host such that each link of G N is stretched to a path in H N . The layout measure is attained by counting the length of paths in H N corresponding to the links in G N and with a complexity of finding the best possible function overall graph embedding. This measure can be computed by summing the minimum congestions on each link of H N , called the congestion lemma. In the current study, we discuss and characterize the congestion lemma by considering the regularity and optimality of the guest network. The exact values of the layout are generally hard to find and were known for very restricted combinations of guest and host networks. In this series, we derive the correct layout measures of circulant networks by embedding them into the path- and cycle-of-complete graphs.

Layout of Embedding Circulant Networks into Linear Hexagons and Phenylenes

2013 ◽  
Vol 14 (03) ◽  
pp. 1350010
Author(s):  
INDRA RAJASINGH ◽  
MICHEAL AROCKIARAJ
Keyword(s):  
Interconnection Networks ◽  
Vlsi Design ◽  
Graph Embedding ◽  
Chemical Compounds ◽  
Theoretical Chemistry ◽  
Telecommunication Networks ◽  
Benzenoid Hydrocarbons ◽  
Graph Representations ◽  
Host Graph ◽  
Circulant Networks

Circulant network has been used for decades in the design of computer and telecommunication networks due to optimal fault-tolerance and routing capabilities. Further, it has been used in VLSI design and distributed computation. Hexagonal chains are of great importance of theoretical chemistry because they are the natural graph representations of benzenoid hydrocarbons, a great deal of investigations in mathematical chemistry has been developed to hexagonal chains. Hexagonal chains are exclusively constructed by hexagons of length one. Phenylenes are a class of chemical compounds in which carbon atoms form 6 and 4 membered cycles. Graph embedding has been known as a powerful tool for implementation of parallel algorithms or simulation of different interconnection networks. An embedding f of a guest graph G into a host graph H is a bijection on the vertices such that each edge of G is mapped into a path of H. The wirelength (layout) of this embedding is defined to be the sum of the length of the paths corresponding to the edges of G. In this paper we obtain the minimum wirelength of embedding circulant networks into linear hexagonal chains and linear phenylenes. Further we discuss the embedding of faulty circulant networks into linear hexagonal chains and linear phenylenes.

scholarly journals A Construction for the Hat Problem on a Directed Graph

10.37236/1994 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Rani Hod ◽  
Marcin Krzywkowski
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Maximum Clique ◽  
Clique Number ◽  
Bipartite Graphs ◽  
The Other ◽  
Complete Graphs ◽  
Undirected Graphs ◽  
Asymptotically Optimal

A team of $n$ players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom he is connected by an arc. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The team aims to maximize the probability of a win, and this maximum is called the hat number of the graph.Previous works focused on the hat problem on complete graphs and on undirected graphs. Some cases were solved, e.g., complete graphs of certain orders, trees, cycles, and bipartite graphs. These led Uriel Feige to conjecture that the hat number of any graph is equal to the hat number of its maximum clique.We show that the conjecture does not hold for directed graphs. Moreover, for every value of the maximum clique size, we provide a tight characterization of the range of possible values of the hat number. We construct families of directed graphs with a fixed clique number the hat number of which is asymptotically optimal. We also determine the hat number of tournaments to be one half.

scholarly journals ON PLANARITY OF DIRECT PRODUCT OF MULTIPARTITE COMPLETE GRAPHS

2009 ◽  
Vol 01 (01) ◽  
pp. 85-104 ◽  
Author(s):  
LAURENT BEAUDOU ◽  
PAUL DORBEC ◽  
SYLVAIN GRAVIER ◽  
PRANAVA K. JHA
Keyword(s):  
Direct Product ◽  
Complete Graphs ◽  
The Past

The planarity of the direct product of two graphs has been widely studied in the past. Surprisingly, the missing part is the product with K2, which seems to be less predictible. In this piece of work, we characterize which subdivisions of multipartite complete graphs, have their direct product with K2 planar. This can be seen as a step towards the characterization of all such graphs.

scholarly journals Cops, robbers, and barricades

2021 ◽  
Author(s):  
Erin Kathleen McKenna Meger
Keyword(s):  
Complete Graphs ◽  
New Variant ◽  
Cops And Robbers ◽  
Minimum Number ◽  
Compare And Contrast

Cops, Robbers, and Barricades is a new variant of the game on graphs, Cops and Robbers. In this variant, the robber may build barricades that restrict the movements of the cops. The minimum number of cops required to capture the robber on a graph G is called the barricade-cop number, denoted cB(G). If cB(G) = 1, then G is called barricade-cop-win. The game can be generalized so that the robber may build b(k)-many barricades on vertices during her kth turn, in accordance with barricade rules that dictate the permissible positions of these barricades. The barricade-cop number is determined exactly for complete graphs, cycles, and paths, and we provide bounds on trees and locally-path-like graphs. We compare and contrast variants on the barricade rules, and give an algorithmic characterization of barricade-cop-win graphs with any set of barricade rules.

scholarly journals Cops, robbers, and barricades

2021 ◽  
Author(s):  
Erin Kathleen McKenna Meger
Keyword(s):  
Complete Graphs ◽  
New Variant ◽  
Cops And Robbers ◽  
Minimum Number ◽  
Compare And Contrast

Cops, Robbers, and Barricades is a new variant of the game on graphs, Cops and Robbers. In this variant, the robber may build barricades that restrict the movements of the cops. The minimum number of cops required to capture the robber on a graph G is called the barricade-cop number, denoted cB(G). If cB(G) = 1, then G is called barricade-cop-win. The game can be generalized so that the robber may build b(k)-many barricades on vertices during her kth turn, in accordance with barricade rules that dictate the permissible positions of these barricades. The barricade-cop number is determined exactly for complete graphs, cycles, and paths, and we provide bounds on trees and locally-path-like graphs. We compare and contrast variants on the barricade rules, and give an algorithmic characterization of barricade-cop-win graphs with any set of barricade rules.

On the Line Degree Splitting Graph of a Graph

2017 ◽  
Vol 18 ◽  
pp. 1-10
Author(s):  
B. Basavanagoud ◽  
Roopa S. Kusugal
Keyword(s):  
Bipartite Graphs ◽  
Complete Graphs ◽  
Complete Bipartite Graphs ◽  
Splitting Graph

In this paper, we introduce the concept of the line degree splitting graph of a graph. We obtain some properties of this graph. We find the girth of the line degree splitting graphs. Further, we establish the characterization of graphs whose line degree splitting graphs are eulerian, complete bipartite graphs and complete graphs.

scholarly journals Characterization of edge-colored complete graphs with properly colored Hamilton paths

2006 ◽  
Vol 53 (4) ◽  
pp. 333-346 ◽  
Author(s):  
Jinfeng Feng ◽  
Hans-Erik Giesen ◽  
Yubao Guo ◽  
Gregory Gutin ◽  
Tommy Jensen ◽  
...  
Keyword(s):  
Complete Graphs ◽  
Hamilton Paths
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