Distance-Preserving Subgraphs of Interval Graphs

We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs that have k terminal vertices. We show that every interval graph admits a distance-preserving subgraph with O(k log k) branching vertices. We also prove a matching lower bound by exhibiting an interval graph based on bit-reversal permutation matrices. In addition, we show that interval graphs admit subgraphs with O(k) branching vertices that approximate distances up to an additive term of +1

Recognizing Geometric Intersection Graphs Stabbed by a Line

In this paper, we determine the computational complexity of recognizing two graph classes, \emph{grounded L}-graphs and \emph{stabbable grid intersection} graphs. An L-shape is made by joining the bottom end-point of a vertical ($\vert$) segment to the left end-point of a horizontal ($-$) segment. The top end-point of the vertical segment is known as the {\em anchor} of the L-shape. Grounded L-graphs are the intersection graphs of L-shapes such that all the L-shapes' anchors lie on the same horizontal line. We show that recognizing grounded L-graphs is NP-complete. This answers an open question asked by Jel{\'\i}nek \& T{\"o}pfer (Electron. J. Comb., 2019). Grid intersection graphs are the intersection graphs of axis-parallel line segments in which two vertical (similarly, two horizontal) segments cannot intersect. We say that a (not necessarily axis-parallel) straight line $\ell$ stabs a segment $s$, if $s$ intersects $\ell$. A graph $G$ is a stabbable grid intersection graph ($StabGIG$) if there is a grid intersection representation of $G$ in which the same line stabs all its segments. We show that recognizing $StabGIG$ graphs is $NP$-complete, even on a restricted class of graphs. This answers an open question asked by Chaplick \etal (\textsc{O}rder, 2018).Comment: 18 pages, 11 Figure

Monotone Classes Beyond VNP

We study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat '08, Koiran-Perifel '09, Malod '11, Mahajan-Rao '13) that is believed to be larger than VNP. We show an exponential separation between the monotone version of Poizat's definition, and monotone VNP. We also show that unlike their non-monotone counterparts, these monotone analogues are not equivalent, with exponential separations in some cases. The primary motivation behind our work is to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrube\v{s} and Yehudayoff (2021). In that context, we are able to show that transparent polynomials of large sparsity are hard for the monotone analogues of all known definitions of VPSPACE, except for the one due to Poizat.Comment: 26 pages; the draft has been shortened and simplified to now focus solely on monotone classes beyond VN

Reconfiguring Shortest Paths in Graphs

Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time, so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) revamping road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c) and (d) are restrictions to different graph classes. We show that (a) is intractable, even for relaxed variants of the problem. For (b), (c) and (d), we present efficient algorithms to solve the respective problems. We also generalise the problem to when at most k (for some k >= 2) contiguous vertices on a shortest path can be changed at a time