A Study on Integer Additive Set-Valuations of Signed Graphs

Let $\N$ denote the set of all non-negative integers and \cP(\N) be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function f:V(G)\to \cP(\N)-\{\emptyset\} such that the induced function f^+:E(G) \to \cP(\N)-\{\emptyset\} is defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. A graph which admits an IASL is usually called an IASL-graph. An IASL $f$ of a graph $G$ is said to be an integer additive set-indexer (IASI) of $G$ if the associated function $f^+$ is also injective. In this paper, we define the notion of integer additive set-labeling of signed graphs and discuss certain properties of signed graphs which admits certain types of integer additive set-labelings.Comment: 12 pages, Carpathian Mathematical Publications, Vol. 8, Issue 2, 2015, 12 page

Approximation algorithms for maximum cut with limited unbalance

AbstractWe consider the problem of partitioning the vertices of a weighted graph into two sets of sizes that differ at most by a given threshold B, so as to maximize the weight of the crossing edges. For B equal to 0 this problem is known as Max Bisection, whereas for B equal to the number n of nodes it is the maximum cut problem. We present polynomial time randomized approximation algorithms with non trivial performance guarantees for its solution. The approximation results are obtained by extending the methodology used by Y. Ye for Max Bisection and by combining this technique with another one that uses the algorithm of Goemans and Williamson for the maximum cut problem. When B is equal to zero the approximation ratio achieved coincides with the one obtained by Y. Ye; otherwise it is always above this value and tends to the value obtained by Goemans and Williamson as B approaches the number n of nodes

Une méthode d'énumération des cycles négatifs d'un graphe signé

RésuméUn graphe signé est un graphe non-orienté èdont les arêtes sont positives ou négatives. Un sous-graphe quelconque sera nommé négatif si il contient un nombre impair d'arêtes négatives. Nous avons élaboré une méthode d'énumération des sous-graphes négatifs d'une famille quelconque des sour-graphes d'un graphe signé. À l'aide de cette méthode nous avons déterminé - dans le cas d'un graphe complet signé quelconque - le nombre des k-cycles négatifs, des k-chaînes négatives et aussi quelques propriétés de divisibilité. Ainsi, pour tout graphe complet signé à n sommets le nombre des k-cycles négatifs (3 ⩽ k ⩽ n) est divisible per 2k−2-[log2k-1 et le nombre des chaines négatives à k sommets (2 ⩽ k ⩽ n) est divisible par 2k−1-[log2k. Ces évaluations sont les meilleures possibles

SOME NEW RESULTS ON INTEGER ADDITIVE SET-VALUED SIGNED GRAPHS

International audienceLet X denotes a set of non-negative integers and P(X) be its power set. An integer additive set-labeling (IASL) of a graph G is an injective set-valued function f : V (G) → P(X) − {∅} such that the induced function f+ : E(G) → P(X) − {∅} is defined by f+(uv) = f(u) + f(v); ∀ uv ∈ E(G), where f(u) + f(v) is the sumset of f(u) and f(v). An IASL of a signed graph is an IASL of its underlying graph G together with the signature σ defined by σ(uv) = (−1)|f+(uv)|; ∀ uv ∈ E(Σ). In this paper, we discuss certain characteristics of the signed graphs which admits certain types of integer additive set-labelings

On certain associated graphs of set-valued signed graphs

International audienceLet X be a non-empty set and let Σ be a signed graph, with corresponding underlying graph G and the signature σ. An injective function f : V (Σ) → P(X) is said to be a set-labeling of Σ if f is a set-labeling of the underlying graph G and the signature of Σ is defined by σ (uv) = (−1) | f (u)⊕ f (v)|. A signed graph Σ together with a set-labeling f is known as a set-labeled signed graph and is denoted by Σ f. In this paper, we discuss the characteristics of certain signed graphs associated with given set-valued signed graphs

Visualizing Structural Balance in Signed Networks

Network visualization has established as a key complement to network analysis since the large variety of existing network layouts are able to graphically highlight different properties of networks. However, signed networks, i.e., networks whose edges are labeled as friendly (positive) or antagonistic (negative), are target of few of such layouts and none, to our knowledge, is able to show structural balance, i.e., the tendency of cycles towards including an even number of negative edges, which is a well-known theory for studying friction and polarization. In this work we present Structural-balance-viz: a novel visualization method showing whether a connected signed network is balanced or not and, in the latter case, how close the network is to be balanced. Structural-balance-viz exploits spectral computations of the signed Laplacian matrix to place network's nodes in a Cartesian coordinate system resembling a balance (a scale). Moreover, it uses edge coloring and bundling to distinguish positive and negative interactions. The proposed visualization method has characteristics desirable in a variety of network analysis tasks: Structural-balance-viz is able to provide indications of balance/polarization of the whole network and of each node, to identify two factions of nodes on the basis of their polarization, and to show their cumulative characteristics. Moreover, the layout is reproducible and easy to compare. Structural-balance-viz is validated over synthetic-generated networks and applied to a real-world dataset about political debates confirming that it is able to provide meaningful interpretations