The Vulcan game of Kal-toh: Finding or making triconnected planar subgraphs

Abstract

In the game of Kal-toh depicted in the television series Star Trek: Voyager, players attempt to create convex polyhedra by adding to a jumbled collection of metal rods. Inspired by this fictional game, we formulate graph-theoretical questions about polyhedral subgraphs, i.e., subgraphs that are triconnected and planar. The problem of determining the existence of a polyhedral subgraph within a graph G is shown to be NP-complete, and we also give some non-trivial upper bounds for the problem of determining the minimum number of edge additions necessary to guarantee the existence of a polyhedral subgraph in G. © 2012 Springer-Verlag Berlin Heidelberg.