scholarly journals Induced label graphoidal graphs

2014 ◽  
Vol 6 (2) ◽  
pp. 178-189
Author(s):  
Ismail Sahul Hamid ◽  
Mayamma Joseph
Keyword(s):  
Directed Graph ◽  
Undirected Graph ◽  
Path Decomposition ◽  
Induced Path ◽  
Decomposition Number

Abstract Let G be a non-trivial, simple, finite, connected and undirected graph of order n and size m. An induced acyclic graphoidal decomposition (IAGD) of G is a collection ψ of non-trivial and internally disjoint induced paths in G such that each edge of G lies in exactly one path of ψ. For a labeling f : V → {1, 2, 3, . . . ,n}, let ↑ Gf be the directed graph obtained by orienting the edges uv of G from u to v, provided f(u) < f(v). If the set ψf of all maximal directed induced paths in ↑ Gf with directions ignored is an induced path decomposition of G, then f is called an induced graphoidal labeling of G and G is called an induced label graphoidal graph. The number ηil = min{|ψf| : f is an induced graphoidal labeling of G} is called the induced label graphoidal decomposition number of G. In this paper we introduce and study the concept of induced graphoidal labeling as an extension of graphoidal labeling.

Two Streamlined Depth-First Search Algorithms

1986 ◽  
Vol 9 (1) ◽  
pp. 85-94
Author(s):  
Robert Endre Tarjan
Keyword(s):  
Directed Graph ◽  
Graph Algorithms ◽  
Undirected Graph ◽  
Linear Time ◽  
Search Algorithms ◽  
Depth First Search ◽  
Time Graph

Many linear-time graph algorithms using depth-first search have been invented. We propose simplified versions of two such algorithms, for computing a bipolar orientation or st-numbering of an undirected graph and for finding all feedback vertices of a directed graph.

Undirected Graphs Realizable as Graphs of Modular Lattices

1965 ◽  
Vol 17 ◽  
pp. 923-932 ◽  
Author(s):  
Laurence R. Alvarez
Keyword(s):  
Distributive Lattice ◽  
Partial Order ◽  
Directed Graph ◽  
Undirected Graph ◽  
Single Edge ◽  
Undirected Graphs ◽  
Modular Lattices

If (L, ≥) is a lattice or partial order we may think of its Hesse diagram as a directed graph, G, containing the single edge E(c, d) if and only if c covers d in (L, ≥). This graph we shall call the graph of (L, ≥). Strictly speaking it is the basis graph of (L, ≥) with the loops at each vertex removed; see (3, p. 170).We shall say that an undirected graph Gu can be realized as the graph of a (modular) (distributive) lattice if and only if there is some (modular) (distributive) lattice whose graph has Gu as its associated undirected graph.

scholarly journals Skew Spectra of Oriented Graphs

10.37236/270 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bryan Shader ◽  
Wasin So
Keyword(s):  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Oriented Graphs ◽  
Underlying Graph ◽  
Adjacency Spectrum ◽  
The Relationship ◽  
Skew Adjacency Matrix

An oriented graph $G^{\sigma}$ is a simple undirected graph $G$ with an orientation $\sigma$, which assigns to each edge a direction so that $G^{\sigma}$ becomes a directed graph. $G$ is called the underlying graph of $G^{\sigma}$, and we denote by $Sp(G)$ the adjacency spectrum of $G$. Skew-adjacency matrix $S( G^{\sigma} )$ of $G^{\sigma}$ is introduced, and its spectrum $Sp_S( G^{\sigma} )$ is called the skew-spectrum of $G^{\sigma}$. The relationship between $Sp_S( G^{\sigma} )$ and $Sp(G)$ is studied. In particular, we prove that (i) $Sp_S( G^{\sigma} ) = {\bf i} Sp(G)$ for some orientation $\sigma$ if and only if $G$ is bipartite, (ii) $Sp_S(G^{\sigma}) = {\bf i} Sp(G)$ for any orientation $\sigma$ if and only if $G$ is a forest, where ${\bf i}=\sqrt{-1}$.

scholarly journals Cliques and colorings in generalized Paley graphs and an approach to synchronization

2015 ◽  
Vol 14 (06) ◽  
pp. 1550088 ◽  
Author(s):  
Csaba Schneider ◽  
Ana C. Silva
Keyword(s):  
Finite Field ◽  
Directed Graph ◽  
Undirected Graph ◽  
Multiplicative Group ◽  
Permutation Groups ◽  
Chromatic Numbers ◽  
Paley Graph ◽  
Paley Graphs

Given a finite field, one can form a directed graph using the field elements as vertices and connecting two vertices if their difference lies in a fixed subgroup of the multiplicative group. If -1 is contained in this fixed subgroup, then we obtain an undirected graph that is referred to as a generalized Paley graph. In this paper, we study generalized Paley graphs whose clique and chromatic numbers coincide and link this theory to the study of the synchronization property in 1-dimensional primitive affine permutation groups.

scholarly journals The average covering tree value for directed graph games

2019 ◽  
Vol 39 (2) ◽  
pp. 315-333 ◽  
Author(s):  
Anna Khmelnitskaya ◽  
Özer Selçuk ◽  
Dolf Talman
Keyword(s):  
Directed Graph ◽  
Undirected Graph ◽  
Solution Concept ◽  
Transferable Utility ◽  
Communication Structure ◽  
Gravity Center ◽  
Dominance Relations ◽  
Graph Games ◽  
Covering Tree ◽  
Transferable Utility Games

Abstract We introduce a single-valued solution concept, the so-called average covering tree value, for the class of transferable utility games with limited communication structure represented by a directed graph. The solution is the average of the marginal contribution vectors corresponding to all covering trees of the directed graph. The covering trees of a directed graph are those (rooted) trees on the set of players that preserve the dominance relations between the players prescribed by the directed graph. The average covering tree value is component efficient, and under a particular convexity-type condition it is stable. For transferable utility games with complete communication structure the average covering tree value equals to the Shapley value of the game. If the graph is the directed analog of an undirected graph the average covering tree value coincides with the gravity center solution.

Research of Key Nodes of Botnet Based on P2P

2011 ◽  
Vol 88-89 ◽  
pp. 386-390
Author(s):  
Jian Gao ◽  
Kang Feng Zheng ◽  
Yi Xian Yang ◽  
Xin Xin Niu
Keyword(s):  
Directed Graph ◽  
Undirected Graph ◽  
Detection Algorithm ◽  
P2p Network ◽  
P2p Networks ◽  
Cut Vertex ◽  
Peer Network ◽  
Simulation Results ◽  
Peer To Peer Network ◽  
Key Nodes

The paper applies the segmentation of peer-to- peer network to the defense process of P2P-based botnet, in order to cause the greatest damage on the P2P network. A lot of papers have been researching how to find the key nodes in P2P networks. To solve this problem, this paper proposes distributed detection algorithm NEI and centralized detection algorithm COR for detecting cut vertex, NEI algorithm not only apply to detect cut vertex of directed graph but also to the undirected graph. COR algorithm can reduce the additional communication. Then, this paper carries out simulation on P2P botnet, the simulation results show that the maximum damage on the botnet can be achieved by destructing key nodes.

scholarly journals Intrinsically knotted and 4-linked directed graphs

2018 ◽  
Vol 27 (06) ◽  
pp. 1850037 ◽  
Author(s):  
Thomas Fleming ◽  
Joel Foisy
Keyword(s):  
Directed Graph ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Edge Contraction ◽  
Special Cases ◽  
Linkless Embedding ◽  
The Relationship ◽  
Knots And Links

We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically 3-linked and 4-linked directed graphs. We introduce two operations, consistent edge contraction and H-cyclic subcontraction, as special cases of minors for digraphs, and show that the property of having a linkless embedding is closed under these operations. We analyze the relationship between the number of distinct knots and links in an undirected graph [Formula: see text] and its corresponding symmetric digraph [Formula: see text]. Finally, we note that the maximum number of edges for a graph that is not intrinsically linked is [Formula: see text] in the undirected case, but [Formula: see text] for directed graphs.

CAYLEY SUM GRAPHS OF IDEALS OF A COMMUTATIVE RING

2014 ◽  
Vol 96 (3) ◽  
pp. 289-302 ◽  
Author(s):  
M. AFKHAMI ◽  
Z. BARATI ◽  
K. KHASHYARMANESH ◽  
N. PAKNEJAD
Keyword(s):  
Commutative Ring ◽  
Directed Graph ◽  
Undirected Graph ◽  
Clique Number ◽  
Basic Properties ◽  
Vertex Set ◽  
Ring Graph ◽  
Genus One ◽  
Cayley Sum Graph

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}R$ be a commutative ring, $I(R)$ be the set of all ideals of $R$ and $S$ be a subset of $I^*(R)=I(R)\setminus \{0\}$. We define a Cayley sum digraph of ideals of $R$, denoted by $\overrightarrow{\mathrm{Cay}}^+ (I(R),S)$, as a directed graph whose vertex set is the set $I(R)$ and, for every two distinct vertices $I$ and $J$, there is an arc from $I$ to $J$, denoted by $I\longrightarrow J$, whenever $I+K=J$, for some ideal $K $ in $S$. Also, the Cayley sum graph $ \mathrm{Cay}^+ (I(R), S)$ is an undirected graph whose vertex set is the set $I(R)$ and two distinct vertices $I$ and $J$ are adjacent whenever $I+K=J$ or $J+K=I$, for some ideal $K $ in $ S$. In this paper, we study some basic properties of the graphs $\overrightarrow{\mathrm{Cay}}^+ (I(R),S)$ and $ \mathrm{Cay}^+ (I(R), S)$ such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of $ \mathrm{Cay}^+ (I(R), S)$ and also we provide some characterization for rings $R$ whose Cayley sum graphs have genus one.

scholarly journals Decomposition of Graphs into Paths and Cycles

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
S. Arumugam ◽  
I. Sahul Hamid ◽  
V. M. Abraham
Keyword(s):  
Minimum Cardinality ◽  
Path Decomposition ◽  
Edge Disjoint ◽  
The Difference ◽  
Decomposition Number

A decomposition of a graph is a collection of edge-disjoint subgraphs of such that every edge of belongs to exactly one . If each is a path or a cycle in , then is called a path decomposition of . If each is a path in , then is called an acyclic path decomposition of . The minimum cardinality of a path decomposition (acyclic path decomposition) of is called the path decomposition number (acyclic path decomposition number) of and is denoted by () (()). In this paper we initiate a study of the parameter and determine the value of for some standard graphs. Further, we obtain some bounds for and characterize graphs attaining the bounds. We also prove that the difference between the parameters and can be made arbitrarily large.

scholarly journals Induced Graphoidal Decompositions in Product Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Mayamma Joseph ◽  
I. Sahul Hamid
Keyword(s):  
Undirected Graph ◽  
Minimum Cardinality ◽  
Product Graph ◽  
Product Graphs ◽  
Decomposition Number

Let be a nontrivial, simple, finite, connected, and undirected graph. A graphoidal decomposition (GD) of is a collection of nontrivial paths and cycles in that are internally disjoint such that every edge of lies in exactly one member of . By restricting the members of a GD to be induced, the concept of induced graphoidal decomposition (IGD) of a graph has been defined. The minimum cardinality of an IGD of a graph is called the induced graphoidal decomposition number and is denoted by (). An IGD of without any cycles is called an induced acyclic graphoidal decomposition (IAGD) of , and the minimum cardinality of an IAGD of is called the induced acyclic graphoidal decomposition number of , denoted by (). In this paper we determine the value of () and () when is a product graph, the factors being paths/cycles.

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