# More on spanning 2-connected subgraphs of alphabet graphs, special classes of grid graphs

M. Salman, Haitze J. Broersma, C.A. Rodger

Research output: Book/ReportReportOther research output

### Abstract

A grid graph $G$ is a finite induced subgraph of the infinite 2-dimensional grid defined by $Z \times Z$ and all edges between pairs of vertices from $Z \times Z$ at Euclidean distance precisely 1. A natural drawing of $G$ is obtained by drawing its vertices in $\mathbb{R}^2$ according to their coordinates. Apart from the outer face, all (inner) faces with area exceeding one (not bounded by a 4-cycle) in a natural drawing of $G$ are called the holes of $G$. We define 26 classes of grid graphs called alphabet graphs, with no or a few holes. We determine which of the alphabet graphs contain a Hamilton cycle, i.e. a cycle containing all vertices, and solve the problem of determining a spanning 2-connected subgraph with as few edges as possible for all alphabet graphs.
Original language English Enschede University of Twente, Department of Applied Mathematics Published - 2003

### Publication series

Name Memorandum Department of Applied Mathematics, University of Twente 1702 0169-2690

### Fingerprint

Grid Graph
Subgraph
Graph in graph theory
Face
Cycle
Hamilton Cycle
Induced Subgraph
Euclidean Distance
Grid
Class
Drawing

• MSC-05C40
• IR-65887
• EWI-3522
• MSC-05C85

### Cite this

Salman, M., Broersma, H. J., & Rodger, C. A. (2003). More on spanning 2-connected subgraphs of alphabet graphs, special classes of grid graphs. (Memorandum; No. 1702). Enschede: University of Twente, Department of Applied Mathematics.
Salman, M. ; Broersma, Haitze J. ; Rodger, C.A. / More on spanning 2-connected subgraphs of alphabet graphs, special classes of grid graphs. Enschede : University of Twente, Department of Applied Mathematics, 2003. (Memorandum; 1702).
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keywords = "MSC-05C40, IR-65887, EWI-3522, MSC-05C85",
author = "M. Salman and Broersma, {Haitze J.} and C.A. Rodger",
year = "2003",
language = "English",
series = "Memorandum",
publisher = "University of Twente, Department of Applied Mathematics",
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Salman, M, Broersma, HJ & Rodger, CA 2003, More on spanning 2-connected subgraphs of alphabet graphs, special classes of grid graphs. Memorandum, no. 1702, University of Twente, Department of Applied Mathematics, Enschede.
Enschede : University of Twente, Department of Applied Mathematics, 2003. (Memorandum; No. 1702).

Research output: Book/ReportReportOther research output

TY - BOOK

T1 - More on spanning 2-connected subgraphs of alphabet graphs, special classes of grid graphs

AU - Salman, M.

AU - Broersma, Haitze J.

AU - Rodger, C.A.

PY - 2003

Y1 - 2003

N2 - A grid graph $G$ is a finite induced subgraph of the infinite 2-dimensional grid defined by $Z \times Z$ and all edges between pairs of vertices from $Z \times Z$ at Euclidean distance precisely 1. A natural drawing of $G$ is obtained by drawing its vertices in $\mathbb{R}^2$ according to their coordinates. Apart from the outer face, all (inner) faces with area exceeding one (not bounded by a 4-cycle) in a natural drawing of $G$ are called the holes of $G$. We define 26 classes of grid graphs called alphabet graphs, with no or a few holes. We determine which of the alphabet graphs contain a Hamilton cycle, i.e. a cycle containing all vertices, and solve the problem of determining a spanning 2-connected subgraph with as few edges as possible for all alphabet graphs.

AB - A grid graph $G$ is a finite induced subgraph of the infinite 2-dimensional grid defined by $Z \times Z$ and all edges between pairs of vertices from $Z \times Z$ at Euclidean distance precisely 1. A natural drawing of $G$ is obtained by drawing its vertices in $\mathbb{R}^2$ according to their coordinates. Apart from the outer face, all (inner) faces with area exceeding one (not bounded by a 4-cycle) in a natural drawing of $G$ are called the holes of $G$. We define 26 classes of grid graphs called alphabet graphs, with no or a few holes. We determine which of the alphabet graphs contain a Hamilton cycle, i.e. a cycle containing all vertices, and solve the problem of determining a spanning 2-connected subgraph with as few edges as possible for all alphabet graphs.

KW - MSC-05C40

KW - IR-65887

KW - EWI-3522

KW - MSC-05C85

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T3 - Memorandum

BT - More on spanning 2-connected subgraphs of alphabet graphs, special classes of grid graphs

PB - University of Twente, Department of Applied Mathematics

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Salman M, Broersma HJ, Rodger CA. More on spanning 2-connected subgraphs of alphabet graphs, special classes of grid graphs. Enschede: University of Twente, Department of Applied Mathematics, 2003. (Memorandum; 1702).