Grid recognition: Classical and parameterized computational perspectives

Siddharth Gupta, Guy Sa'ar, Meirav Zehavi

Abstract: Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph is hard—it was shown to be NP-hard already in 1987. In this paper, we provide several positive results in this regard in the framework of parameterized complexity. Specifically, our contribution is threefold. First, we show that the problem is FPT parameterized by k+mcc<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mrow is="true"><mi mathvariant="sans-serif" is="true">mcc</mi></mrow></math> where mcc<math><mi mathvariant="sans-serif" is="true">mcc</mi></math> is the maximum size of a connected component of G. Second, we present a new parameterization, denoted aG<math><msub is="true"><mrow is="true"><mi is="true">a</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub></math>, relating graph distance to geometric distance. We show that the problem is para-NP-hard parameterized by aG<math><msub is="true"><mrow is="true"><mi is="true">a</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub></math>, but FPT parameterized by aG<math><msub is="true"><mrow is="true"><mi is="true">a</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub></math> on trees, as well as FPT parameterized by k+aG<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><msub is="true"><mrow is="true"><mi is="true">a</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub></math>. Third, we show that the recognition of k×r<math><mi is="true">k</mi><mo is="true">×</mo><mi is="true">r</mi></math> grid graphs is NP-hard on graphs of pathwidth 2 where k=3<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">3</mn></math>.