We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. For positive we have a construction which shows that all chordal graphs that can be represented as intersection graph of subpaths on a tree are pseudosegment intersection graphs. We then study the limits of representability. We describe a family of intersection graphs of substars of a star which is not representable as intersection graph of pseudosegments. The degree of the substars in this example, however, has to get large. A more intricate analysis involving a Ramsey argument shows that even in the class of intersection graphs of substars of degree three of a star there are graphs that are not representable as intersection graph of pseudosegments.
Motivated by representability questions for chordal graphs we consider how many combinatorially different k-segments, i.e., curves crossing k distinct lines, an arrangement of n pseudolines can host. We show that for fixed k this number is in O(n^2). This result is based on a k-zone theorem for arrangements of pseudolines that should be of independent interest.
Strongly chordal graphs are a subclass of chordal graphs. Farber also established a number of different characterisations for this class of graphs. These include an intersection graph characterisation that is analogous to a similar characterisation for chordal graphs. Seker et al. exploited this characterisation of chordal graphs to obtain an algorithm for generating them. In this paper, we propose an algorithm to show that strongly chordal graphs can also be generated using their intersection graph characterisation.
The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.
The leafage l(G) of a chordal graph G is the minimum number of leaves of a tree in which G has an intersection representation by subtrees. We obtain upper and lower bounds on l(G) and compute it on special classes. The maximum of l(G) on n-vertex graphs is n - lg n - (1/2) lg lg n + O(1). The proper leafage l*(G) is the minimum number of leaves when no subtree may contain another; we obtain upper and lower bounds on l*(G). Leafage equals proper leafage on claw-free chordal graphs. We use asteroidal sets and structural properties of chordal graphs.
The perfect phylogeny problem is a classic problem in computational biology, where we seek an unrooted phylogeny that is compatible with a set of qualitative characters. Such a tree exists precisely when an intersection graph associated with the character set, called the partition intersection graph, can be triangulated using a restricted set of fill edges. Semple and Steel used the partition intersection graph to characterize when a character set has a unique perfect phylogeny. Bordewich, Huber, and Semple showed how to use the partition intersection graph to find a maximum compatible set of …
A universal representation theorem is derived that shows any graph is the intersection graph of one chordal graph, a number of co-bipartite graphs, and one unit interval graph. Central to the the result is the notion of the clique cover width which is a generalization of the bandwidth parameter. Specifically, we show that any planar graph is the intersection graph of one chordal graph, four co-bipartite graphs, and one unit interval graph. Equivalently, any planar graph is the intersection graph of a chordal graph and a graph that has {clique cover width} of at most seven. We further describe …
We investigate graphs that can be represented as vertex intersections of horizontal and vertical paths in a grid, the so called $B_0$-VPG graphs. Recognizing this class is an NP-complete problem. Although, there exists a polynomial time algorithm for recognizing chordal $B_0$-VPG graphs. In this paper, we present a minimal forbidden induced subgraph characterization of $B_0$-VPG graphs restricted to block graphs. As a byproduct, the proof of the main theorem provides an alternative certifying recognition and representation algorithm for $B_0$-VPG graphs in the class of block graphs.
The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a representing tree, and the leafage of a chordal graph is defined to be the minimum number of leaves in a representing tree for it. We prove that chordal graph isomorphism is fixed parameter tractable with leafage as parameter. In the process we intro…
For a family of graphs $\mathcal{F}$, Weighted $\mathcal{F}$-Deletion is the problem for which the input is a vertex weighted graph $G=(V,E)$ and the goal is to delete $S\subseteq V$ with minimum weight such that $G\setminus S\in\mathcal{F}$. Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more…
Let $G$ be a simple undirected graph on $n$ vertices. Francisco and Van Tuyl have shown that if $G$ is chordal, then $\bigcap_{\{x_i,x_j\}\in E_G} < x_i,x_j>$ is componentwise linear. A natural question that arises is for which $t_{ij}>1$ the ideal $\bigcap_{\{x_i,x_j\}\in E_G}< x_i, x_j>^{t_{ij}}$ is componentwise linear, if $G$ is chordal. In this report we show that $\bigcap_{\{x_i,x_j\}\in E_G} < x_i, x_j>^{t}$ is componentwise linear for all $n\geq 3$ and positive $t$, if $G$ is a complete graph. We give also an example where $G$ is chordal, but the intersection ideal is not componentwise…
The class ${\cal L}_k$ of $k$-leaf powers consists of graphs $G=(V,E)$ that have a $k$-leaf root, that is, a tree $T$ with leaf set $V$, where $xy \in E$, if and only if the $T$-distance between $x$ and $y$ is at most $k$. Structure and linear time recognition algorithms have been found for $2$-, $3$-, $4$-, and, to some extent, $5$-leaf powers, and it is known that the union of all $k$-leaf powers, that is, the graph class ${\cal L} = \bigcup_{k=2}^\infty {\cal L}_k$, forms a proper subclass of strongly chordal graphs. Despite from that, no essential progress has been made lately. In this pap…
A directed graph $G=(V,E)$ is {\it strongly pseudo transitive} if there is a partition $\{A,E-A\}$ of $E$ so that graphs $G_1=(V,A)$ and $G_2=(V,E-A)$ are transitive, and additionally, if $ab\in A$ and $bc\in E $ implies that $ac\in E$. A strongly pseudo transitive graph $G=(V,E)$ is strongly pseudo transitive of the first type, if $ab\in A$ and $bc\in E$ implies $ac\in A$. An undirected graph is co-strongly pseudo transitive (co-strongly pseudo transitive of the first type) if its complement has an orientation which is strongly pseudo transitive (co-strongly pseudo transitive of the first typ…
Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is nonempty. We consider a theorem of Hilton which gives a best possible upper bound on the sum of the cardinalities of uniform cross-intersecting subfamilies. We formulate a graph-theoretic analogue of Hilton's cross-intersection theorem, similar to the one developed by Holroyd, Spencer and Talbot for the Erdos-Ko-Rado theorem. In particular we build on a result …
Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. It measures the tree-likeness of a graph from a metric viewpoint. In particular, we are interested in circular-arc graphs, which is an important class of geometric intersection graphs. In this paper we give sharp bounds for the hyperbolicity constant of (finite and infinite) circular-arc graphs. Moreover, we obtain bounds for the hyperbolicity constant of the complement and line of any circular-arc graph. In order to do that, we obtain new results about regular, chord…
It was noted already in the 90s that many classic graph classes, such as interval, chordal, and bipartite graphs, can be characterized by the existence of an ordering of the vertices avoiding some ordered subgraphs, called patterns. Very recently, all the classes corresponding to patterns on three vertices (including the ones mentioned above) have been listed, and proved to be efficiently recognizable. In contrast, very little is known about patterns on four vertices. One of the few graph classes characterized by a pattern on four vertices is the class of intersection graphs of rectangles that…
When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman and Singer-Cohen [Combin Probab Comput 8 (1999), 131--159] introduced a random intersection graph by taking randomly assigned sets. The random intersection graph $G(n,m;p)$ has $n$ …
A new metric parameter for a graph, Helly-gap, is introduced. A graph $G$ is called $α$-weakly-Helly if any system of pairwise intersecting disks in $G$ has a nonempty common intersection when the radius of each disk is increased by an additive value $α$. The minimum $α$ for which a graph $G$ is $α$-weakly-Helly is called the Helly-gap of $G$ and denoted by $α(G)$. The Helly-gap of a graph $G$ is characterized by distances in the injective hull $\mathcal{H}(G)$, which is a (unique) minimal Helly graph which contains $G$ as an isometric subgraph. This characterization is used as a tool to gener…
Biró et al. (1992) introduced $H$-graphs, intersection graphs of connected subgraphs of a subdivision of a graph $H$. They are related to many classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. We negatively answer the 25-year-old question of Biró et al. which asks if $H$-graphs can be recognized in polynomial time, for a fixed graph $H$. We prove that it is NP-complete if $H$ contains the diamond graph as a minor. We provide a polynomial-time algorithm recognizing $T$-graphs, for each fixed tree $T$. When $T$ is a star $S_d$…
