### Abstract

An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by the author to answer various two-dimensional regular lattice model problems in enumerative combinatorics and statistical mechanics.

Original language | English |
---|---|

Pages (from-to) | 2762-2768 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 340 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2017 Dec 1 |

### Fingerprint

### Keywords

- Enumeration
- Grid graph
- Maximal independent set

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*340*(12), 2762-2768. https://doi.org/10.1016/j.disc.2017.08.015

**Maximal independent sets on a grid graph.** / Oh, Seung Sang.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 340, no. 12, pp. 2762-2768. https://doi.org/10.1016/j.disc.2017.08.015

}

TY - JOUR

T1 - Maximal independent sets on a grid graph

AU - Oh, Seung Sang

PY - 2017/12/1

Y1 - 2017/12/1

N2 - An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by the author to answer various two-dimensional regular lattice model problems in enumerative combinatorics and statistical mechanics.

AB - An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by the author to answer various two-dimensional regular lattice model problems in enumerative combinatorics and statistical mechanics.

KW - Enumeration

KW - Grid graph

KW - Maximal independent set

UR - http://www.scopus.com/inward/record.url?scp=85028751960&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028751960&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2017.08.015

DO - 10.1016/j.disc.2017.08.015

M3 - Article

AN - SCOPUS:85028751960

VL - 340

SP - 2762

EP - 2768

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 12

ER -