Intersection Graphs Of Curves Research Articles

A Sharp Threshold Phenomenon in String Graphs

A string graph is the intersection graph of curves in the plane. We prove that for every epsilon >0, if G is a string graph with n vertices such that the edge density of G is below {1}/{4}-epsilon , then V(G) contains two linear sized subsets A and B with no edges between them. The constant 1/4 is a sharp threshold for this phenomenon as there are string graphs with edge density less than {1}/{4}+epsilon such that there is an edge connecting any two logarithmic sized subsets of the vertices. The existence of linear sized sets A and B with no edges between them in sufficiently sparse string graphs is a direct consequence of a recent result of Lee about separators. Our main theorem finds the largest possible density for which this still holds. In the special case when the curves are x-monotone, the same result was proved by Pach and the author of this paper, who also proposed the conjecture for the general case.

Conflict-Free Coloring of String Graphs

Conflict-free coloring (in short, CF-coloring) of a graph $$G = (V,E)$$ is a coloring of V such that the punctured neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex in that neighborhood. Bounds on CF-chromatic numbers have been studied both for general graphs and for intersection graphs of geometric shapes. In this paper we obtain such bounds for several classes of string graphs, i.e., intersection graphs of curves in the plane: (i) we provide a general upper bound of $$O(\chi (G)^2 \log n)$$ on the CF-chromatic number of any string graph G with n vertices in terms of the classical chromatic number $$\chi (G)$$ . This result stands in contrast to general graphs where the CF-chromatic number can be $$\varOmega (\sqrt{n})$$ already for bipartite graphs. (ii) For some central classes of string graphs, the CF-chromatic number is as large as $$\varTheta (\sqrt{n})$$ , which was shown to be the upper bound for any graph even in the non-geometric context. For several such classes (e.g., intersection graphs of frames) we prove a tight bound of $$\varTheta (\log n)$$ with respect to the relaxed notion of k-CF-coloring (in which the punctured neighborhood of each vertex contains a color that appears at most k times), for a small constant k. (iii) We obtain a general upper bound on the k-CF chromatic number of arbitrary hypergraphs (i.e., the number of colors needed to color the vertices, such that in each hyperedge there is a color that appears at most k times): any hypergraph with m hyperedges can be k-CF colored with $${\widetilde{O}}\bigl (m^{\frac{1}{k+1}}\bigr )$$ colors. This bound, which extends a bound of Pach and Tardos (Comb Probab Comput 18(5):819–834, 2009), is tight for some string graphs, up to a logarithmic factor. (iv) Our fourth result concerns circle graphs in which coloring problems are motivated by VLSI designs. We prove a tight bound of $$\varTheta (\log n)$$ on the CF-chromatic number of circle graphs, and an upper bound of $$O(\log ^{3} n)$$ for a wider class of string graphs that contains circle graphs, namely, intersection graphs of grounded L-shapes.

Subexponential algorithms for variants of the homomorphism problem in string graphs

We consider subexponential algorithms finding weighted homomorphisms from intersection graphs of curves (string graphs) with n vertices to a fixed graph H. We provide a complete dichotomy: if H has no two vertices sharing two common neighbors, then the problem can be solved in time 2O(n2/3log⁡n), otherwise there is no subexponential algorithm, assuming the ETH. Then we consider locally constrained homomorphisms. We show that for each target graph H, the locally injective and locally bijective homomorphism problems can be solved in time 2O(nlog⁡n) in string graphs. For locally surjective homomorphisms we show a dichotomy for H being a path or a cycle. If H is P3 or C4, then the problem can be solved in time 2O(n2/3log3/2⁡n) in string graphs; otherwise, assuming the ETH, there is no subexponential algorithm. As corollaries, we obtain new results concerning the complexity of homomorphism problems in Pt-free graphs.

Ordered graphs and large bi-cliques in intersection graphs of curves

An ordered graphG< is a graph with a total ordering < on its vertex set. A monotone path of length k−1 is a sequence of vertices v1<v2<…<vk such that vivj is an edge of G< if and only if |j−i|=1. A bi-clique of size m is a complete bipartite graph whose vertex classes are of size m.We prove that for every positive integer k, there exists a constant ck>0 such that every ordered graph on n vertices that does not contain a monotone path of length k as an induced subgraph has a vertex of degree at least ckn, or its complement has a bi-clique of size at least ckn∕logn. A similar result holds for ordered graphs containing no induced ordered subgraph isomorphic to a fixed ordered matching.As a consequence, we give a short combinatorial proof of the following theorem of Fox and Pach. There exists a constant c>0 such the intersection graph G of any collection of nx-monotone curves in the plane has a bi-clique of size at least cn∕logn or its complement contains a bi-clique of size at least cn. (A curve is called x-monotone if every vertical line intersects it in at most one point.) We also prove that if G has at most 14−ϵn2 edges for some ϵ>0, then G¯ contains a linear sized bi-clique. We show that this statement does not remain true if we replace 14 by any larger constants.

Optimality Program in Segment and String Graphs

Planar graphs are known to allow subexponential algorithms running in time 2^{O(sqrt{n})} or 2^{O(sqrt{n} log n)} for most of the paradigmatic problems, while the brute-force time 2^{varTheta (n)} is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in 2^{O(n^{2/3}log n)} by Fox and Pach (SODA’11), we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the Exponential Time Hypothesis (ETH). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time 2^{O(n^{2/3}log ^{O(1)}n)} on string graphs while an algorithm running in time 2^{o(n)} for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker lower bound, excluding a 2^{o(n^{2/3})} algorithm (under the ETH). The construction exploits the celebrated Erdős–Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set, but not to Min Dominating Set and Min Independent Dominating Set.

Outerstring Graphs are $\chi$-Bounded

An outerstring graph is an intersection graph of curves that lie in a common half-plane and have one endpoint on the boundary of that half-plane. We prove that the class of outerstring graphs is $\...

An algorithm for the maximum weight independent set problem on outerstring graphs

Outerstring graphs are the intersection graphs of curves that lie inside a disk such that each curve intersects the boundary of the disk. Outerstring graphs are among the most general classes of intersection graphs studied. To date, no polynomial time algorithm is known for any of the classical graph optimization problems on outerstring graphs; in fact, most are NP-hard. It is known that there is an intersection model for any outerstring graph that consists of polygonal arcs attached to a circle. However, this representation may require an exponential number of segments relative to the size of the graph.Given an outerstring graph and an intersection model consisting of polygonal arcs with a total of N segments, we develop an algorithm that solves the Maximum Weight Independent Set problem in O(N3) time. If the polygonal arcs are restricted to single segments, then outersegment graphs result. For outersegment graphs, we solve the Maximum Weight Independent Set problem in O(n3) time where n is the number of vertices in the graph.

Applications of a New Separator Theorem for String Graphs

An intersection graph of curves in the plane is called astring graph. Matoušek almost completely settled a conjecture of the authors by showing that every string graph withmedges admits a vertex separator of size$O(\sqrt{m}\log m)$. In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphsGwithnvertices: (i) ifKt⊈Gfor somet, then the chromatic number ofGis at most (logn)O(logt); (ii) ifKt,t⊈G, thenGhas at mostt(logt)O(1)nedges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdős–Hajnal conjecture almost holds for string graphs.

Weakly transitive orientations, Hasse diagrams and string graphs

We introduce the notion of a weakly transitive orientation for graphs as a natural generalization of transitive orientations and give a characterization for weakly transitive orientations in terms of forbidden substructurs. As a corollary of this characterization, we get that the Hasse diagrams of posets are exactly the triangle-free weakly transitive oriented graphs. Moreover, we characterize the complements of weakly transitive orientable graphs as a special class of intersection graphs of paths in planar graphs. In this way, complements of weakly transitive orientable graphs prove to form a common generalization of several known classes of intersection graphs such as circular-permutation graphs (and, thus, circular-arc graphs), complements of comparability graphs (and, thus, permutation graphs and interval graphs). An immediate consequence of this theorem is an intersection graph characterization of Hasse diagrams. Using this characterization, we obtain a simple reduction of the problem to decide for a graph whether it admits an orientation as a Hasse diagram to the recognition problem for string graphs. Using a result of Nešetřil and Rödl that Hasse diagram orientation is NP-complete, this gives a new proof for NP-hardness of the string graph recognition problem. The status of complexity of the recognition problem for string graphs (intersection graphs of curves in the plane) was a long-standing open problem until Kratochvil gave a (rather involved) NP-hardness proof in 1988.

String graphs. I. The number of critical nonstring graphs is infinite

String graphs (intersection graphs of curves in the plane) were originally studied in connection with RC-circuits. The family of string graphs is closed in the induced minor order, and so it is reasonable to study critical nonstring graphs (nonstring graphs such that all of their proper induced minors are string graphs). The question of whether there are infinitely many nonisomorphic critical nonstring graphs has been an open problem for some time. The main result of this paper settles this question. In a later paper of this series we show that recognizing string graphs is NP-hard.

String graphs. II. recognizing string graphs is NP-hard

We prove that recognizing string graphs (i.e., intersection graphs of curves in the plane) and some related problems are NP-hard.