Seidel complementation on (P5P_5, HouseHouse, BullBull)-free graphs

We consider the Seidel complementation on ($P_5, \bar{P_5}, Bull)$-free graph

Characterization and enumeration of 3-regular permutation graphs

A permutation graph is a graph that can be derived from a permutation, where the vertices correspond to letters of the permutation, and the edges represent inversions. We provide a construction to show that there are infinitely many connected $r$-regular permutation graphs for $r \geq 3$. We prove that all $3$-regular permutation graphs arise from a similar construction. Finally, we enumerate all $3$-regular permutation graphs on $n$ vertices.Comment: 11 pages, 11 figure

Comparing rankings by means of competitivity graphs: structural properties and computation

In this paper we introduce a new technique to analyze families of rankings focused on the study of structural properties of a new type of graphs. Given a finite number of elements and a family of rankings of those elements, we say that two elements compete when they exchange their relative positions in at least two rankings. This allows us to define an undirected graph by connecting elements that compete. We call this graph a competitivity graph. We study the relationship of competitivity graphs with other well-known families of graphs, such as permutation graphs, comparability graphs and chordal graphs. In addition to this, we also introduce certain important sets of nodes in a competitivity graph. For example, nodes that compete among them form a competitivity set and nodes connected by chains of competitors form a set of eventual competitors. We analyze hese sets and we show a method to obtain sets of eventual competitors directly from a family of rankings

Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

A graph $G$ with $n$ vertices is called an outerstring graph if it has an intersection representation of a set of $n$ curves inside a disk such that one endpoint of every curve is attached to the boundary of the disk. Given an outerstring graph representation, the Maximum Independent Set (MIS) problem of the underlying graph can be solved in $O(s^3)$ time, where $s$ is the number of segments in the representation (Keil et al., Comput. Geom., 60:19--25, 2017). If the strings are of constant size (e.g., line segments, L-shapes, etc.), then the algorithm takes $O(n^3)$ time. In this paper, we examine the fine-grained complexity of the MIS problem on some well-known outerstring representations. We show that solving the MIS problem on grounded segment and grounded square-L representations is at least as hard as solving MIS on circle graph representations. Note that no $O(n^{2-\delta})$-time algorithm, $\delta>0$, is known for the MIS problem on circle graphs. For the grounded string representations where the strings are $y$-monotone simple polygonal paths of constant length with segments at integral coordinates, we solve MIS in $O(n^2)$ time and show this to be the best possible under the strong exponential time hypothesis (SETH). For the intersection graph of $n$ L-shapes in the plane, we give a $(4\cdot \log OPT)$-approximation algorithm for MIS (where $OPT$ denotes the size of an optimal solution), improving the previously best-known $(4\cdot \log n)$-approximation algorithm of Biedl and Derka (WADS 2017).Comment: 16 pages, 8 figure