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

• Main Author: Anderson, Terry David
• Language: en
• Published: 2011
• Subjects: triconnectivity; planarity; polyhedra; subgraphs; Computer Science
• Online Access: http://hdl.handle.net/10012/5882

In the game of Kal-toh depicted in the television series Star Trek: Voyager, players attempt to create polyhedra by adding to a jumbled collection of metal rods. Inspired by this fictional game, we formulate graph-theoretical questions about polyhedral (triconnected and planar) subgraphs in an on-line environment. The problem of determining the existence of a polyhedral subgraph within a graph G is shown to be NP-hard, 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. A two-player formulation of Kal-toh is also explored, in which the first player to form a target subgraph is declared the winner. We show a polynomial-time solution for simple cases of this game but conjecture that the general problem is NP-hard.