Related papers: On Gracefully Labeling Trees
Assume that we embed the path $P_n$ as a subgraph of a $2$-dimensional grid, namely, $P_k \times P_l$. Given such an embedding, we consider the ordered set of subpaths $L_1, L_2, \ldots , L_m$ which are maximal straight segments in the…
The intersection graph of a collection of trapezoids with corner points lying on two parallel lines is called a trapezoid graph. Using binary indexed tree data structure, we improve algorithms for calculating the size and the number of…
Let $S$ be a set of $n$ points in the plane in general position. Two line segments connecting pairs of points of $S$ cross if they have an interior point in common. Two vertex disjoint geometric graphs with vertices in $S$ cross if there…
In this paper, we provide an algorithm for traversing geometric graphs which visits all vertices, and reports every vertex and edge exactly once. To achieve this, we combine a given geometric graph $G$ with the integer lattice, seen as a…
Laman graphs are fundamental to rigidity theory. A graph G with n vertices and m edges is a generic minimally rigid graph (Laman graph), if m=2n-3 and every induced subset of k vertices spans at most 2k-3 edges. We consider the verification…
Given a graph $G(V, E)$ and a positive integer $k$ ($k \geq 1$), a simple path on $k$ vertices is a sequence of $k$ vertices in which no vertex appears more than once and each consecutive pair of vertices in the sequence are connected by an…
In the first part of this paper I give an elementary overview about some number sequences which count various sorts of lattice paths in strips along the x-axis and compute their generating functions in terms of Fibonacci and Lucas…
Perfect codes in the $n$-dimensio\-nal grid $\Lambda_n$ of the lattice $\mathbb{Z}^n$ ($0<n\in\mathbb{Z}$) and its quotient toroidal grids were obtained via the truncated distance in $\mathbb{Z}^n$ given between $u=(u_1,\cdots,u_n)$ and…
We consider questions related to the existence of spanning trees in graphs with the property that after the removal of any path in the tree the graph remains connected. We show that, for planar graphs, the existence of trees with this…
A connected set in a graph is a non-empty set of vertices that induces a connected subgraph. In an infinite lattice, a connected set is often referred to as a lattice animal, whose enumeration up to isomorphism is a classical problem in…
Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is a collection of edge-disjoint paths that covers all the edges of $G$. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of $G$. Gallai…
We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight…
Consider the Birkhoff polytope of n by n doubly-stochastic matrices. As the Birkhoff-von Neumann theorem famously states, its vertex set coincides with the set of all n by n permutation matrices. Here we seek a higher-dimensional analog of…
Euler graphs are characterized by the simple criterion that degree of each node is even. By restricting on the cycle types yet additional intrinsic properties of Euler graphs are unveiled. For example, regularity higher than degree two is…
Let $\Gamma$ be a finite, undirected, connected, simple graph. We say that a matching $\mathcal{M}$ is a \textit{permutable $m$-matching} if $\mathcal{M}$ contains $m$ edges and the subgroup of $\text{Aut}(\Gamma)$ that fixes the matching…
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…
Graphs provide an efficient tool for object representation in various computer vision applications. Once graph-based representations are constructed, an important question is how to compare graphs. This problem is often formulated as a…
The Disjoint Paths Problem asks, given a graph $G$ and a set of pairs of terminals $(s_{1},t_{1}),\ldots,(s_{k},t_{k})$, whether there is a collection of $k$ pairwise vertex-disjoint paths linking $s_{i}$ and $t_{i}$, for $i=1,\ldots,k.$ In…
The lattice dimension of a graph G is the minimal dimension of a cubic lattice in which G can be isometrically embedded. We prove that the lattice dimension of a tree with n leaves is $\lceil n/2 \rceil$.
Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in…