TY - JOUR

T1 - Maximal independent sets on a grid graph

AU - Oh, Seungsang

N1 - Funding Information:
This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (No. NRF-2014R1A2A1A11050999 ).
Publisher Copyright:
© 2017 Elsevier B.V.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2017/12

Y1 - 2017/12

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

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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 -