EPG-representations with small grid-size

In an EPG-representation of a graph $G$ each vertex is represented by a path in the rectangular grid, and $(v,w)$ is an edge in $G$ if and only if the paths representing $v$ an $w$ share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x- and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters. We show that there are $m$-edge graphs that require a grid of area $\Omega(m)$ in any variant of EPG-representations. Similarly there are pathwidth-$k$ graphs that require height $\Omega(k)$ and area $\Omega(kn)$ in any variant of EPG-representations. We prove a matching upper bound of $O(kn)$ area for all pathwidth-$k$ graphs in the strongest model, the one where edges are required to be both x- and y-monotone. Thus in this strongest model, the result implies, for example, $O(n)$, $O(n \log n)$ and $O(n^{3/2})$ area bounds for bounded pathwidth graphs, bounded treewidth graphs and all classes of graphs that exclude a fixed minor, respectively. For the model with no restrictions on the monotonicity of the edges, stronger results can be achieved for some graph classes, for example an $O(n)$ area bound for bounded treewidth graphs and $O(n \log^2 n)$ bound for graphs of bounded genus.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Planar graphs as L-intersection or L-contact graphs

The L-intersection graphs are the graphs that have a representation as intersection graphs of axis parallel shapes in the plane. A subfamily of these graphs are {L, |, --}-contact graphs which are the contact graphs of axis parallel L, |, and -- shapes in the plane. We prove here two results that were conjectured by Chaplick and Ueckerdt in 2013. We show that planar graphs are L-intersection graphs, and that triangle-free planar graphs are {L, |, --}-contact graphs. These results are obtained by a new and simple decomposition technique for 4-connected triangulations. Our results also provide a much simpler proof of the known fact that planar graphs are segment intersection graphs

Proper circular arc graphs as intersection graphs of paths on a grid

In this paper we present a characterisation, by an infinite family of minimal forbidden induced subgraphs, of proper circular arc graphs which are intersection graphs of paths on a grid, where each path has at most one bend (turn)

Relationship of kk-Bend and Monotonic ℓ\ell-Bend Edge Intersection Graphs of Paths on a Grid

If a graph $G$ can be represented by means of paths on a grid, such that each vertex of $G$ corresponds to one path on the grid and two vertices of $G$ are adjacent if and only if the corresponding paths share a grid edge, then this graph is called EPG and the representation is called EPG representation. A $k$-bend EPG representation is an EPG representation in which each path has at most $k$ bends. The class of all graphs that have a $k$-bend EPG representation is denoted by $B_k$. $B_\ell^m$ is the class of all graphs that have a monotonic (each path is ascending in both columns and rows) $\ell$-bend EPG representation. It is known that $B_k^m \subsetneqq B_k$ holds for $k=1$. We prove that $B_k^m \subsetneqq B_k$ holds also for $k \in \{2, 3, 5\}$ and for $k \geqslant 7$ by investigating the $B_k$-membership and $B_k^m$-membership of complete bipartite graphs. In particular we derive necessary conditions for this membership that have to be fulfilled by $m$, $n$ and $k$, where $m$ and $n$ are the number of vertices on the two partition classes of the bipartite graph. We conjecture that $B_{k}^{m} \subsetneqq B_{k}$ holds also for $k\in \{4,6\}$. Furthermore we show that $B_k \not\subseteq B_{2k-9}^m$ holds for all $k\geqslant 5$. This implies that restricting the shape of the paths can lead to a significant increase of the number of bends needed in an EPG representation. So far no bounds on the amount of that increase were known. We prove that $B_1 \subseteq B_3^m$ holds, providing the first result of this kind

Restricted String Representations

A string representation of a graph assigns to every vertex a curve in the plane so that two curves intersect if and only if the represented vertices are adjacent. This work investigates string representations of graphs with an emphasis on the shapes of curves and the way they intersect. We strengthen some previously known results and show that every planar graph has string representations where every curve consists of axis-parallel line segments with at most two bends (those are the so-called $B_2$-VPG representations) and simultaneously two curves intersect each other at most once (those are the so-called 1-string representations). Thus, planar graphs are $B_2$-VPG $1$-string graphs. We further show that with some restrictions on the shapes of the curves, string representations can be used to produce approximation algorithms for several hard problems. The $B_2$-VPG representations of planar graphs satisfy these restrictions. We attempt to further restrict the number of bends in VPG representations for subclasses of planar graphs, and investigate $B_1$-VPG representations. We propose new classes of string representations for planar graphs that we call ``order-preserving.'' Order-preservation is an interesting property which relates the string representation to the planar embedding of the graph, and we believe that it might prove useful when constructing string representations. Finally, we extend our investigation of string representations to string representations that require some curves to intersect multiple times. We show that there are outer-string graphs that require an exponential number of crossings in their outer-string representations. Our construction also proves that 1-planar graphs, i.e., graphs that are no longer planar, yet fairly close to planar graphs, may have string representations, but they are not always 1-string

Computing maximum cliques in B2B_2-EPG graphs

EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection graphs of paths on an orthogonal grid. The class $B_k$-EPG is the subclass of EPG graphs where the path on the grid associated to each vertex has at most $k$ bends. Epstein et al. showed in 2013 that computing a maximum clique in $B_1$-EPG graphs is polynomial. As remarked in [Heldt et al., 2014], when the number of bends is at least $4$, the class contains $2$-interval graphs for which computing a maximum clique is an NP-hard problem. The complexity status of the Maximum Clique problem remains open for $B_2$ and $B_3$-EPG graphs. In this paper, we show that we can compute a maximum clique in polynomial time in $B_2$-EPG graphs given a representation of the graph. Moreover, we show that a simple counting argument provides a ${2(k+1)}$-approximation for the coloring problem on $B_k$-EPG graphs without knowing the representation of the graph. It generalizes a result of [Epstein et al, 2013] on $B_1$-EPG graphs (where the representation was needed)

Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid

A graph $G$ is called an edge intersection graph of paths on a grid if there is a grid and there is a set of paths on this grid, such that the vertices of $G$ correspond to the paths and two vertices of $G$ are adjacent if and only if the corresponding paths share a grid edge. Such a representation is called an EPG representation of $G$. $B_{k}$ is the class of graphs for which there exists an EPG representation where every path has at most $k$ bends. The bend number $b(G)$ of a graph $G$ is the smallest natural number $k$ for which $G$ belongs to $B_k$. $B_{k}^{m}$ is the subclass of $B_k$ containing all graphs for which there exists an EPG representation where every path has at most $k$ bends and is monotonic, i.e. it is ascending in both columns and rows. The monotonic bend number $b^m(G)$ of a graph $G$ is the smallest natural number $k$ for which $G$ belongs to $B_k^m$. Edge intersection graphs of paths on a grid were introduced by Golumbic, Lipshteyn and Stern in 2009 and a lot of research has been done on them since then. In this paper we deal with the monotonic bend number of outerplanar graphs. We show that $b^m(G)\leqslant 2$ holds for every outerplanar graph $G$. Moreover, we characterize in terms of forbidden subgraphs the maximal outerplanar graphs and the cacti with (monotonic) bend number equal to $0$, $1$ and $2$. As a consequence we show that for any maximal outerplanar graph and any cactus a (monotonic) EPG representation with the smallest possible number of bends can be constructed in a time which is polynomial in the number of vertices of the graph

Characterising circular-arc contact B0B_0-VPG graphs

A contact $B_0$-VPG graph is a graph for which there exists a collection of nontrivial pairwise interiorly disjoint horizontal and vertical segments in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding segments touch. It was shown by Deniz et al. that Recognition is $\mathsf{NP}$-complete for contact $B_0$-VPG graphs. In this paper we present a minimal forbidden induced subgraph characterisation of contact $B_0$-VPG graphs within the class of circular-arc graphs and provide a polynomial-time algorithm for recognising these graphs

Proper circular arc graphs as intersection graphs of pathson a grid

In this paper we present a characterization, by an infinite family of minimal forbidden induced subgraphs, of proper circular arc graphs which are intersection graphs of paths on a grid, where each path has at most one bend (turn).Facultad de Ciencias Exacta