Splitting cubic circle graphs

We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition. We also deduce that up to isomorphism, K_4 and K_{3,3} are the only 3-connected, 3-regular circle graphs.Comment: 18 pages, 15 figure

Filtering graphs to check isomorphism and extracting mapping by using the Conductance Electrical Model

© 2016. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper presents a new method of filtering graphs to check exact graph isomorphism and extracting their mapping. Each graph is modeled by a resistive electrical circuit using the Conductance Electrical Model (CEM). By using this model, a necessary condition to check the isomorphism of two graphs is that their equivalent resistances have the same values, but this is not enough, and we have to look for their mapping to find the sufficient condition. We can compute the isomorphism between two graphs in O(N-3), where N is the order of the graph, if their star resistance values are different, otherwise the computational time is exponential, but only with respect to the number of repeated star resistance values, which usually is very small. We can use this technique to filter graphs that are not isomorphic and in case that they are, we can obtain their node mapping. A distinguishing feature over other methods is that, even if there exists repeated star resistance values, we can extract a partial node mapping (of all the nodes except the repeated ones and their neighbors) in O(N-3). The paper presents the method and its application to detect isomorphic graphs in two well know graph databases, where some graphs have more than 600 nodes. (C) 2016 Elsevier Ltd. All rights reserved.Postprint (author's draft

Isomorphism of graph classes related to the circular-ones property

We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and convex-round graphs.Comment: 25 pages, 9 figure

Computing NodeTrix Representations of Clustered Graphs

NodeTrix representations are a popular way to visualize clustered graphs; they represent clusters as adjacency matrices and inter-cluster edges as curves connecting the matrix boundaries. We study the complexity of constructing NodeTrix representations focusing on planarity testing problems, and we show several NP-completeness results and some polynomial-time algorithms. Building on such algorithms we develop a JavaScript library for NodeTrix representations aimed at reducing the crossings between edges incident to the same matrix.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016