scholarly journals On the 12-representability of induced subgraphs of a grid graph

2019 ◽  
Author(s):  
Joanna N. Chen ◽  
Sergey Kitaev
Keyword(s):  
Grid Graph ◽  
Induced Subgraphs

scholarly journals Representing Graphs via Pattern Avoiding Words

10.37236/4946 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Miles Jones ◽  
Sergey Kitaev ◽  
Artem Pyatkin ◽  
Jeffrey Remmel
Keyword(s):  
Finite Graph ◽  
Interval Graphs ◽  
Comparability Graph ◽  
Grid Graph ◽  
Induced Subgraphs ◽  
Permutation Graphs

The notion of a word-representable graph has been studied in a series of papers in the literature. A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. If $V =\{1, \ldots, n\}$, this is equivalent to saying that $G$ is word-representable if for all $x,y \in \{1, \ldots, n\}$, $xy \in E$ if and only if the subword $w_{\{x,y\}}$ of $w$ consisting of all occurrences of $x$ or $y$ in $w$ has no consecutive occurrence of the pattern 11.In this paper, we introduce the study of $u$-representable graphs for any word $u \in \{1,2\}^*$. A graph $G$ is $u$-representable if and only if there is a vertex-labeled version of $G$, $G=(\{1, \ldots, n\}, E)$, and a word $w \in \{1, \ldots, n\}^*$ such that for all $x,y \in \{1, \ldots, n\}$, $xy \in E$ if and only if $w_{\{x,y\}}$ has no consecutive occurrence of the pattern $u$. Thus, word-representable graphs are just $11$-representable graphs. We show that for any $k \geq 3$, every finite graph $G$ is $1^k$-representable. This contrasts with the fact that not all graphs are 11-representable graphs.The main focus of the paper is the study of $12$-representable graphs. In particular, we classify the $12$-representable trees. We show that any $12$-representable graph is a comparability graph and the class of $12$-representable graphs include the classes of co-interval graphs and permutation graphs. We also state a number of facts on $12$-representation of induced subgraphs of a grid graph.

Excluding induced subgraphs: Critical graphs

2010 ◽  
Vol 38 (1-2) ◽  
pp. 100-120 ◽  
Author(s):  
József Balogh ◽  
Jane Butterfield
Keyword(s):  
Induced Subgraphs ◽  
Critical Graphs

Grid Graph Reachability Problems

2006 ◽  
Author(s):  
E. Allender ◽  
D.A. Mix Barrington ◽  
T. Chakraborty ◽  
S. Datta ◽  
S. Roy
Keyword(s):  
Grid Graph ◽  
Graph Reachability

scholarly journals Degree and neighborhood intersection conditions restricted to induced subgraphs ensuring Hamiltonicity of graphs

2014 ◽  
Vol 06 (03) ◽  
pp. 1450043
Author(s):  
Bo Ning ◽  
Shenggui Zhang ◽  
Bing Chen
Keyword(s):  
Hamilton Cycles ◽  
Induced Subgraphs ◽  
Degree Sum ◽  
Degree Condition

Let claw be the graph K1,3. A graph G on n ≥ 3 vertices is called o-heavy if each induced claw of G has a pair of end-vertices with degree sum at least n, and called 1-heavy if at least one end-vertex of each induced claw of G has degree at least n/2. In this note, we show that every 2-connected o-heavy or 3-connected 1-heavy graph is Hamiltonian if we restrict Fan-type degree condition or neighborhood intersection condition to certain pairs of vertices in some small induced subgraphs of the graph. Our results improve or extend previous results of Broersma et al., Chen et al., Fan, Goodman and Hedetniemi, Gould and Jacobson, and Shi on the existence of Hamilton cycles in graphs.

scholarly journals Maximal independent sets on a grid graph

2017 ◽  
Vol 340 (12) ◽  
pp. 2762-2768 ◽  
Author(s):  
Seungsang Oh
Keyword(s):  
Independent Sets ◽  
Grid Graph ◽  
Maximal Independent Sets

Grid graph planning and control method applied to equipment maintenance in the chemical industry

1967 ◽  
Vol 3 (3) ◽  
pp. 231-234 ◽  
Author(s):  
E. K. Sashina ◽  
�. I. Shklovskii ◽  
A. B. Miller ◽  
Yu. S. Chentsov
Keyword(s):  
Chemical Industry ◽  
Control Method ◽  
Equipment Maintenance ◽  
Grid Graph ◽  
Planning And Control ◽  
And Control ◽  
Graph Planning
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