Hamiltonian Paths in Two Classes of Grid Graphs

In this paper, we give the necessary and sufficient conditions for the existence of Hamiltonian paths in $L-$alphabet and $C-$alphabet grid graphs. We also present a linear-time algorithm for finding Hamiltonian paths in these graphs.Comment: 11pages, 7figure

Unit-length embedding of cycles and paths on grid graphs

Although there are very algorithms for embedding graphs on unbounded grids, only few results on embedding or drawing graphs on restricted grids has been published. In this work, we consider the problem of embedding paths and cycles on grid graphs. We give the necessary and sufficient conditions for the existence of cycles of given length $k$ and paths of given length $k$ between two given vertices in $n$-vertex rectangular grid graphs and introduce two algorithms with running times O$(k)$ and O$(k^2)$ for finding respectively such cycles and paths. Also, we extend our results to $m\times n\times o$ 3D grids. Our method for finding cycle of length $k$ in rectangular grid graphs also introduces a linear-time algorithm for finding cycles of a given length $k$ in hamiltonian solid grid graphs

A Framework for ETH-Tight Algorithms and Lower Bounds in Geometric Intersection Graphs

We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to a wide range of geometric intersection graphs (intersections of similarly sized fat objects), yielding algorithms with running time $2^{O(n^{1-1/d})}$ for any fixed dimension $d \geq 2$ for many well known graph problems, including Independent Set, $r$-Dominating Set for constant $r$, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms work on the graph itself, i.e., do not require any geometric information. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques. The lower bound framework is based on a constructive embedding of graphs into d-dimensional grids, and it allows us to derive matching $2^{\Omega(n^{1-1/d})}$ lower bounds under the Exponential Time Hypothesis even in the much more restricted class of $d$-dimensional induced grid graphs.Comment: 37 pages, full version of STOC 2018 paper; v2 updates the title and fixes some typo

Hamiltonian Cycles in Linear-Convex Supergrid Graphs

A supergrid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional supergrid. The supergrid graphs contain grid graphs and triangular grid graphs as subgraphs. The Hamiltonian cycle problem for grid and triangular grid graphs was known to be NP-complete. In the past, we have shown that the Hamiltonian cycle problem for supergrid graphs is also NP-complete. The Hamiltonian cycle problem on supergrid graphs can be applied to control the stitching trace of computerized sewing machines. In this paper, we will study the Hamiltonian cycle property of linear-convex supergrid graphs which form a subclass of supergrid graphs. A connected graph is called $k$-connected if there are $k$ vertex-disjoint paths between every pair of vertices, and is called locally connected if the neighbors of each vertex in it form a connected subgraph. In this paper, we first show that any 2-connected, linear-convex supergrid graph is locally connected. We then prove that any 2-connected, linear-convex supergrid graph contains a Hamiltonian cycle.Comment: 17 pages, 24 figur

Hamiltonian Paths in C-shaped Grid Graphs

We study the Hamiltonian path problem in C-shaped grid graphs, and present the necessary and sufficient conditions for the existence of a Hamiltonian path between two given vertices in these graphs. We also give a linear-time algorithm for finding a Hamiltonian path between two given vertices of a C-shaped grid graph, if it exists.Comment: 28 pages, 31 figures, and 20 reference

Computing and Sampling Restricted Vertex Degree Subgraphs and Hamiltonian Cycles

Let $G=(V,E)$ be a bipartite graph embedded in a plane (or $n$-holed torus). Two subgraphs of $G$ differ by a {\it $Z$-transformation} if their symmetric difference consists of the boundary edges of a single face---and if each subgraph contains an alternating set of the edges of that face. For a given $\phi: V \mapsto \mathbb Z^+$, $S_{\phi}$ is the set of subgraphs of $G$ in which each $v\in V$ has degree $\phi(v)$. Two elements of $S_{\phi}$ are said to be adjacent if they differ by a $Z$-transformation. We determine the connected components of $S_{\phi}$ and assign a {\it height function} to each of its elements. If $\phi$ is identically two, and $G$ is a grid graph, $S_{\phi}$ contains the partitions of the vertices of $G$ into cycles. We prove that we can always apply a series of $Z$-transformations to decrease the total number of cycles provided there is enough ``slack'' in the corresponding height function. This allows us to determine in polynomial time the minimal number of cycles into which $G$ can be partitioned provided $G$ has a limited number of non-square faces. In particular, we determine the Hamiltonicity of polyomino graphs in $O(|V|^2)$ steps. The algorithm extends to $n$-holed-torus-embedded graphs that have grid-like properties. We also provide Markov chains for sampling and approximately counting the Hamiltonian cycles of $G$.Comment: 42 pages, fifteen figures, includes new reference

The Hamiltonicity, Hamiltonian Connectivity, and Longest (s, t)-path of L-shaped Supergrid Graphs

Supergrid graphs contain grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian cycle and path problems for general supergrid graphs were known to be NP-complete. A graph is called Hamiltonian if it contains a Hamiltonian cycle, and is said to be Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices in it. In this paper, we first prove that every L-shaped supergrid graph always contains a Hamiltonian cycle except one trivial condition. We then verify the Hamiltonian connectivity of L-shaped supergrid graphs except few conditions. The Hamiltonicity and Hamiltonian connectivity of L-shaped supergrid graphs can be applied to compute the minimum trace of computerized embroidery machine and 3D printer when a L-like object is printed. Finally, we present a linear-time algorithm to compute the longest (s, t)-path of L-shaped supergrid graph given two distinct vertices s and t.Comment: A preliminary version of this paper has appeared in: The International MultiConference of Engineers and Computer Scientists 2018 (IMECS 2018), Hong Kong, vol. I, 2018, pp. 117-12

Simultaneous Embedding of Planar Graphs

Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic.Comment: survey, 35 pages, 12 figure

2-Trees: Structural Insights and the study of Hamiltonian Paths

For a connected graph, a path containing all vertices is known as \emph{Hamiltonian path}. For general graphs, there is no known necessary and sufficient condition for the existence of Hamiltonian paths and the complexity of finding a Hamiltonian path in general graphs is NP-Complete. We present a necessary and sufficient condition for the existence of Hamiltonian paths in 2-trees. Using our characterization, we also present a linear-time algorithm for the existence of Hamiltonian paths in 2-trees. Our characterization is based on a deep understanding of the structure of 2-trees and the combinatorics presented here may be used in other combinatorial problems restricted to 2-trees.Comment: 16 pages, 7 figure

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

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