Related papers: Boxicity of Halin Graphs

A $k$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$…

The boxicity of a graph $G$, denoted by $box(G)$, is the least positive integer $\ell$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $\ell$-space, where box in an Euclidean $\ell$-space is the…

The 'boxicity' ('cubicity') of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in $R^k$. In this article, we give estimates on…

A Halin graph is a graph obtained by embedding a tree having no nodes of degree two in the plane, and then adding a cycle to join the leaves of the tree in such a way that the resulting graph is planar. According to the four color theorem,…

The boxicity of a graph $G=(V,E)$ is the least integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap ... \cap E_k$. Scheinerman proved in 1984 that outerplanar graphs have boxicity at…

Halin graphs constitute an interesting class of planar and polyhedral graphs. A generalized Halin graph is obtained by connecting all leaves of a planar embedding of a tree via a cycle. A Halin graph is a generalized Halin graph having no…

The boxicity of a graph $G=(V,E)$ is the smallest integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap \cdots \cap E_k$. In the first part of this note, we prove that every graph on $m$…

A box in Euclidean $k$-space is the Cartesian product of $k$ closed intervals on the real line. The boxicity of a graph $G$, denoted by $\text{box}(G)$, is the minimum nonnegative integer $k$ such that $G$ can be isomorphic to the…

Let $T$ be a tree with no degree 2 vertices and $L(T)=\{l_1,\ldots,l_r\}, r \geq 2$ denote the set of leaves in $T$. An Halin graph $G$ is a graph obtained from $T$ such that $V(G)=V(T)$ and $E(G)=E(T) \cup \{\{l_i,l_{i+1}\} ~|~ 1 \leq i…

A box in Euclidean $k$-space is the Cartesian product $I_1\times I_2\times \cdots \times I_k$, where $I_j$ is a closed interval on the real line. The boxicity of a graph $G$, denoted by $\text{box}(G)$, is the minimum nonnegative integer…

Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $\delta(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there…

For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least \[…

An axis parallel $d$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer…

The \emph{local boxicity} of a graph $G$, denoted by $lbox(G)$, is the minimum positive integer $l$ such that $G$ can be obtained using the intersection of $k$ (, where $k \geq l$,) interval graphs where each vertex of $G$ appears as a…

Boxicity of a graph $G(V,$ $E)$, denoted by $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of axis parallel boxes in $\mathbb{R}^k$. The problem of computing boxicity is inapproximable even…

It is known that if G is a connected simple graph, then G^3 is Hamiltonian (in fact, Hamilton-connected). A simple graph is k-ordered Hamiltonian if for any sequence v_1, v_2, ..., v_k of k vertices there is a Hamiltonian cycle containing…

A box is the cartesian product of real intervals, which are either bounded or equal to $\mathbb{R}$. A box is said to be $d$-local if at most $d$ of the intervals are bounded. In this paper, we investigate the recently introduced local…

The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative integer $k$, such that $G$ can be represented as an intersection graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit cubes) and is…

Let $G$ be a simple graph of order $n$ and let $k$ be an integer such that $1\leq k\leq n-1$. The $k$-token graph $G^{\{k\}}$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $G^{\{k\}}$…

Let $k$ and $n$ be integers such that $1\leq k \leq n-1$, and let $G$ be a simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$…