[Paper Review] The maximum forcing number of polyomino

Liqiong Xu, Yuqing Lin+1 more|arXiv (Cornell University)|Oct 3, 2014

Advanced Graph Theory Research10 references3 citations

TL;DR

This paper proves that the maximum forcing number of an elementary polyomino equals its Clar number, enabling polynomial-time computation of the maximum forcing number for such graphs. The result extends to non-elementary polyominoes and hexagonal systems by decomposing them into elementary components and fixed bonds, establishing that the maximum forcing number can be computed in polynomial time for all bipartite, 2-connected, planar graphs with perfect matchings.

ABSTRACT

The forcing number of a perfect matching $M$ of a graph $G$ is the cardinality of the smallest subset of $M$ that is contained in no other perfect matchings of $G$. For a planar embedding of a 2-connected bipartite planar graph $G$ which has a perfect matching, the concept of Clar number of hexagonal system had been extended by Abeledo and Atkinson as follows: a spanning subgraph $C$ of is called a Clar cover of $G$ if each of its components is either an even face or an edge, the maximum number of even faces in Clar covers of $G$ is called Clar number of $G$, and the Clar cover with the maximum number of even faces is called the maximum Clar cover. It was proved that if $G$ is a hexagonal system with a perfect matching $M$ and $K'$ is a set of hexagons in a maximum Clar cover of $G$, then $G-K'$ has a unique 1-factor. Using this result, Xu {\it et. at.} proved that the maximum forcing number of the elementary hexagonal system are equal to their Clar numbers, and then the maximum forcing number of the elementary hexagonal system can be computed in polynomial time. In this paper, we show that an elementary polyomino has a unique perfect matching when removing the set of tetragons from its maximum Clar cover. Thus the maximum forcing number of elementary polyomino equals to its Clar number and can be computed in polynomial time. Also, we have extended our result to the non-elementary polyomino and hexagonal system.

Motivation & Objective

• To resolve the conjecture that the maximum forcing number of an elementary polyomino can be computed in polynomial time.
• To establish a connection between the maximum forcing number and Clar number in polyominoes.
• To generalize the polynomial-time computability of the maximum forcing number to non-elementary polyominoes and hexagonal systems.
• To provide a computational framework based on decomposition into elementary components and fixed bonds.

Proposed method

• Prove that removing all tetragons from a maximum Clar cover of an elementary polyomino results in a graph with a unique perfect matching.
• Use this uniqueness to show that the maximum forcing number of an elementary polyomino equals its Clar number.
• Leverage existing algorithms for computing Clar numbers via linear programming to compute the maximum forcing number in polynomial time.
• Apply decomposition techniques to non-elementary polyominoes and hexagonal systems into elementary components and fixed bonds.
• Use the fact that the maximum forcing number of a non-elementary graph is the sum of the maximum forcing numbers of its elementary components.
• Utilize an O(|E| + |V|) algorithm to identify elementary components and fixed bonds in bipartite graphs with perfect matchings.

Experimental results

Research questions

• RQ1 Does the maximum forcing number of an elementary polyomino equal its Clar number?
• RQ2 Can the maximum forcing number of an elementary polyomino be computed in polynomial time?
• RQ3 Does the equality between maximum forcing number and Clar number extend to non-elementary polyominoes and hexagonal systems?
• RQ4 What is the computational complexity of computing the maximum forcing number for non-elementary polyominoes and hexagonal systems?

Key findings

• The maximum forcing number of an elementary polyomino equals its Clar number.
• The maximum forcing number of an elementary polyomino can be computed in polynomial time.
• For non-elementary polyominoes and hexagonal systems, the maximum forcing number equals the sum of the maximum forcing numbers of their elementary components.
• The decomposition of non-elementary graphs into elementary components and fixed bonds can be performed in O(|E| + |V|) time.
• The maximum forcing number of any polyomino or hexagonal system with a perfect matching can be computed in polynomial time.
• The result generalizes prior findings on elementary hexagonal systems to all bipartite, 2-connected, planar graphs with perfect matchings.