Abstract
Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves (2011) initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space {{mathbb {R}}}^{l} in such a manner that every vertex is closer to all its friends (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, embeddability into {{mathbb {R}}}^{1} can be expressed as a purely combinatorial problem. In this paper we pursue a deeper study of this case, answering several questions posed by Kermarrec and Thraves. First, we refine the approach of Kermarrec and Thraves for the case of complete signed graphs by showing that the problem is closely related to the recognition of proper interval graphs. Second, we prove that the general case, whose polynomial-time tractability remained open, is in fact NP-complete. Finally, we provide lower and upper bounds for the time complexity of the general case: we prove that the existence of a subexponential time (in the number of vertices and edges of the input signed graph) algorithm would violate the Exponential Time Hypothesis, whereas a simple dynamic programming approach gives a running time single-exponential in the number of vertices.
Highlights
Undirected graphs with edges labelled positively and negatively, called signed graphs, in many applications serve as a very simple model of relationships between a group of people, e.g., in a social network
Given a signed graph G, is it possible to embed the vertices of G in Rl so that for any positive edge uu1 and negative edge uu2 it holds that d(u, u1) < d(u, u2)? This question has a natural interpretation: we would like to place a group of people so that every person is placed closer to his friends than to his enemies
Our Results We focus on the problem of embedding a signed graph into a line
Summary
Undirected graphs with edges labelled positively (by a +) and negatively (by a −), called signed graphs, in many applications serve as a very simple model of relationships between a group of people, e.g., in a social network. One of the problems is to visualize the model graph properly, i.e., in such a way that positive relations tend to make vertices be placed close to each other, while negative relations imply large distances between vertices In their recent work, Kermarrec and Thraves [14] formalized this problem as follows: Consider the metric space Rl with the Euclidean metric denoted by d. Kermarrec and Thraves posed a number of open problems in the area, including the question of the complexity of determining the embeddability of an arbitrary (not necessarily complete) graph into the Euclidean line. We resolve the open problem posed in [14] negatively: it is NP -complete to resolve whether a given signed graph can be embedded into a line This hardness result answers other questions of Kermarrec and Thraves [14].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
