Related papers: The lattice dimension of a tree

The mixed metric dimension ${\rm mdim}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that (metrically) resolves each pair of elements from $V(G)\cup E(G)$. We say that $G$ is a max-mdim graph if ${\rm mdim}(G) = n(G)$.…

A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph $G$ is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of $G$. We…

We prove that if a graph has a tree-decomposition of width at most w, then it has a tree-decomposition of width at most w with certain desirable properties. We will use this result in a subsequent paper to show that every 2-connected graph…

The Metric Dimension problem asks for a minimum-sized resolving set in a given (unweighted, undirected) graph $G$. Here, a set $S \subseteq V(G)$ is resolving if no two distinct vertices of $G$ have the same distance vector to $S$. The…

For a vertex $u$ of a tree $T$, the leaf (internal, respectively) status of $u$ is the sum of the distances from $u$ to all leaves (internal vertices, respectively) of $T$. The minimum (maximum, respectively) leaf status of a tree $T$ is…

We give a constructive proof of the fact that the treewidth of a graph $G$ is bounded by a linear function of the separation number of $G$.

A graph G is a 2-tree if G=K_3, or G has a vertex v of degree 2, whose neighbours are adjacent, and G\v{i}s a 2-tree. A characterization of the degree sequences of 2-trees is given. This characterization yields a linear-time algorithm for…

The unit-distance graph on the $n$-dimensional integer lattice $\mathbb{Z}^n$ is called the $n$-dimensional grid. We attempt to maximize the girth of a $k$-regular (possibly induced) subgraph of the $n$-dimensional grid, and provide…

For a tree $T$, let $lp(T)$ be the number of different lengths of leaf to leaf paths in $T$. For a degree sequence $s$ of a tree, let ${\rm rad}(s)$ be the minimum radius of a tree with degree sequence $s$. Recently, Di Braccio,…

A set of vertices $S$ of a graph $G$ is a (geodesic)convex set, if $S$ contains all the vertices belonging to any shortest path connecting between two vertices of $S$. The cardinality of maximum proper convex set of $G$ is called the…

A graph $G=(V,E)$ is geometrically embeddable into a normed space $X$ when there is a mapping $\zeta: V\to X$ such that $\|\zeta(v)-\zeta(w)\|_X\leqslant 1$ if and only if $\{v,w\}\in E$, for all distinct $v,w\in V$. Our result is the…

We study the $SIG$ dimension of trees under $L_{\infty}$ metric and answer an open problem posed by Michael and Quint (Discrete Applied Mathematics: 127, pages 447-460, 2003). Let $T$ be a tree with atleast two vertices. For each $v\in…

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all $s\geq 1$, every connected graph on $n$ vertices with minimum degree at least $(\frac{1}{s+3}+o(1))n$ contains a spanning tree having at most $s$ branch…

Non-rigidity degree of a lattice $L$, nrd$L$, is dimension of the L-type domain to which $L$ belongs. We complete here the table of nrd's of all root lattices and their duals; namely, the hardest remaining case of $D_n^*$, and the case of…

In this note, we give answers to three questions from the paper [A. Das, Triameter of graphs, Discuss. Math. Graph Theory, 41 (2021), 601--616]. Namely, we obtain a tight lower bound for the triameter of trees in terms of order and number…

We prove that for every graph $G$ with $n$ vertices, the treewidth of $G$ plus the treewidth of the complement of $G$ is at least $n-2$. This bound is tight.

Contraction$^*$-depth is a matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph $G$ is the minimum height of a rooted…

By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of 'grid-likeness' of a graph. The grid contains a long…

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the vertices $i$ and $j$ in $G$. We consider a weighted tree $T$ on $n$ vertices with edge weights are square matrix of…

Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut dimension of $G$ is the dimension of the span of the characteristic vectors of the minimum cuts of $G$, viewed as vectors in $\{0,1\}^m$. For every $n \ge 2$ we show…