Characterizing Simultaneous Embeddings with Fixed Edges

A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same Jordan curve in the simultaneous drawings. While any number of planar graphs have a simultaneous embedding without fixed edges, determining which graphs always share a simultaneous embedding with fixed edges (SEFE) has been open. We partially close this problem by giving a necessary condition to determine when pairs of graphs have a SEFE

Characterizing Simultaneous Embeddings with Fixed Edges

A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same Jordan curve in the simultaneous drawings. While any number of planar graphs have a simultaneous embedding without ?xed edges, determining which graphs always share a simultaneous embedding with ?xed edges (SEFE) has been open. We partially close this problem by giving a necessary condition to determine when pairs of graphs have a SEFE

Characterizations of Restricted Pairs of Planar Graphs allowing Simultaneous Embeddings with Fixed Edges

A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with ?xed edges (SEFE) has been open. We give a necessary and su?cient condition for whether a SEFE exists for pairs of graphs whose union is homeomorphic to K5 or K3,3 . This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide e?cient algorithms to compute a SEFE. Finally, we provide a linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has a SEFE

Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs

We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a finite list of minimal forbidden induced subgraphs. These, therefore, disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming that all such minimal classes must be defined by infinitely many forbidden induced subgraphs. In the same paper, Daligault, Rao and Thomasse make another conjecture that every hereditary class of unbounded clique-width must contain a labelled infinite antichain. We show that the two example classes we consider here satisfy this conjecture. Indeed, they each contain a canonical labelled infinite antichain, which leads us to propose a stronger conjecture: that every hereditary class of graphs that is minimal of unbounded clique-width contains a canonical labelled infinite antichain.Comment: 17 pages, 7 figure

Revisiting Resolution and Inter-Layer Coupling Factors in Modularity for Multilayer Networks

Modularity for multilayer networks, also called multislice modularity, is parametric to a resolution factor and an inter-layer coupling factor. The former is useful to express layer-specific relevance and the latter quantifies the strength of node linkage across the layers of a network. However, such parameters can be set arbitrarily, thus discarding any structure information at graph or community level. Other issues are related to the inability of properly modeling order relations over the layers, which is required for dynamic networks. In this paper we propose a new definition of modularity for multilayer networks that aims to overcome major issues of existing multislice modularity. We revise the role and semantics of the layer-specific resolution and inter-layer coupling terms, and define parameter-free unsupervised approaches for their computation, by using information from the within-layer and inter-layer structures of the communities. Moreover, our formulation of multilayer modularity is general enough to account for an available ordering of the layers and relating constraints on layer coupling. Experimental evaluation was conducted using three state-of-the-art methods for multilayer community detection and nine real-world multilayer networks. Results have shown the significance of our modularity, disclosing the effects of different combinations of the resolution and inter-layer coupling functions. This work can pave the way for the development of new optimization methods for discovering community structures in multilayer networks.Comment: Accepted at the IEEE/ACM Conf. on Advances in Social Network Analysis and Mining (ASONAM 2017