Abstract
We introduce the notion of stab number and exact stab number of rectangle intersection graphs, otherwise known as graphs of boxicity at most 2. A graph G is said to be a k-stabbable rectangle intersection graph, or k-SRIG for short, if it has a rectangle intersection representation in which k horizontal lines can be chosen such that each rectangle is intersected by at least one of them. If there exists such a representation with the additional property that each rectangle intersects exactly one of the k horizontal lines, then the graph G is said to be a k-exactly stabbable rectangle intersection graph, or k-ESRIG for short. The stab number of a graph G, denoted by stab(G), is the minimum integer k such that G is a k-SRIG. Similarly, the exact stab number of a graph G, denoted by estab(G), is the minimum integer k such that G is a k-ESRIG. In this work, we study the stab number and exact stab number of some subclasses of rectangle intersection graphs. A lower bound on the stab number of rectangle intersection graphs in terms of its pathwidth and clique number is shown. Tight upper bounds on the exact stab number of split graphs with boxicity at most 2 and block graphs are also given. We show that for k ≤ 3, k-SRIG is equivalent to k-ESRIG and for any k ≥ 10, there is a tree which is a k-SRIG but not a k-ESRIG. We also develop a forbidden structure characterization for block graphs that are 2-ESRIG and trees that are 3-ESRIG, which lead to polynomial-time recognition algorithms for these two classes of graphs. These forbidden structures are natural generalizations of asteroidal triples. Finally, we construct examples to show that these forbidden structures are not sufficient to characterize block graphs that are 3-SRIG or trees that are k-SRIG for any k ≥ 4.
Similar content being viewed by others
References
Adiga, A., Bhowmick, D., Chandran, L.S.: The hardness of approximating the boxicity, cubicity and threshold dimension of a graph. Discret. Appl. Math. 158 (16), 1719–1726 (2010)
Adiga, A., Chandran, L.S., Sivadasan, N.: Lower bounds for boxicity. Combinatorica 34(6), 631–655 (2014)
Agarwal, P.K., Van Kreveld, M., Suri, S.: Label placement by maximum independent set in rectangles. Comput. Geom. 11(3-4), 209–218 (1998)
Asplund, E., Grünbaum, B.: On a coloring problem. Mathematica Scandinavica 8(1), 181–188 (1960)
Babu, J., Basavaraju, M., Chandran, L.S., Rajendraprasad, D., Sivadasan, N.: Approximating the cubicity of trees. arXiv:1402.6310 (2014)
Bhore, S.K., Chakraborty, D., Das, S., Sen, S.: On a Special Class of Boxicity 2 Graphs. In: Algorithms and Discrete Applied Mathematics: First International Conference, pp. 157–168 (2015)
Chan, T.M.: A note on maximum independent sets in rectangle intersection graphs. Inf. Process. Lett. 89(1), 19–23 (2004)
Chandran, L.S., Francis, M.C., Sivadasan, N.: Boxicity and maximum degree. J. Comb. Theory, Ser. B 98(2), 443–445 (2008)
Chandran, L.S., Mathew, R., Rajendraprasad, D.: Upper bound on cubicity in terms of boxicity for graphs of low chromatic number. Discret. Math. 339(2), 443–446 (2016)
Chandran, L.S., Sivadasan, N.: Boxicity and treewidth. J Comb Theory Ser B 97(5), 733–744 (2007)
Corneil, D.G., Olariu, S., Stewart, L.: The LBFS structure and recognition of interval graphs. SIAM J. Discret. Math. 23(4), 1905–1953 (2009)
Correa, J.R., Feuilloley, L., Soto, J.A.: Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line. In: Latin American Symposium on Theoretical Informatics, pp. 35–46. Springer (2014)
Cozzens, M.B., Roberts, F.S.: Computing the boxicity of a graph by covering its complement by cointerval graphs. Discret. Appl. Math. 6(3), 217–228 (1983)
Diestel, R.: Graph Theory. Electronic library of mathematics. Springer, Berlin (2006). https://books.google.co.in/books?id=aR2TMYQr2CMC
Ellis, J., Warren, R.: Lower bounds on the pathwidth of some grid-like graphs. Discret. Appl. Math. 156(5), 545–555 (2008)
Erlebach, T., Van Leeuwen, E.J.: PTAS for Weighted Set Cover on Unit Squares. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 166–177. Springer (2010)
Esperet, L., Joret, G.: Boxicity of graphs on surfaces. Graphs and Combinatorics, 1–11 (2013)
Imai, H., Asano, T.: Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. Algorithm. 4(4), 310–323 (1983)
Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discret. Appl. Math. 52(3), 233–252 (1994)
Kratochvíl, J., Nešetřil, J.: Independent set and clique problems in intersection-defined classes of graphs. Comment. Math. Univ. Carolinae 31(1), 85–93 (1990)
Lekkerkerker, C., Boland, J.: Representation of a finite graph by a set of intervals on the real line. Fundam. Math. 51(1), 45–64 (1962)
Suderman, M.: Pathwidth and layered drawings of trees. Int. J. Comput. Geom. Appl. 14(03), 203–225 (2004)
Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Algebr. Discret. Methods 3(3), 351–358 (1982)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Partially supported by the DST-INSPIRE Faculty Award IFA12-ENG-21.
Rights and permissions
About this article
Cite this article
Chakraborty, D., Francis, M.C. On the Stab Number of Rectangle Intersection Graphs. Theory Comput Syst 64, 681–734 (2020). https://doi.org/10.1007/s00224-019-09936-w
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-019-09936-w