arXiv subjects
Publications and source records attributed to Peter Dankelmann.
It was conjectured by Koh and Tay [Graphs Combin. 18(4) (2002), 745--756] that for $n\geq 5$ every simple graph of order $n$ and size at least $\binom{n}{2}-n+5$ has an orientation of diameter two. We prove this conjecture and hence determine for every $n\geq 5$ the minimum value of $m$ such that every graph of order $n$ and size $m$ has an orientation of diameter two.
The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order $n$ of a graph in terms of its diameter $d$ and metric dimension $k$. In general, the bound $n\leq d^k+k$ is known to hold. We prove a bound of the form $n=\mathcal{O}(kd^2)$ for trees and outerplanar graphs (for trees we determine the best possible bound and the corresponding extremal examples). More generally, for graphs having a tree decomposition of width $w$ and length $\ell$, we obtain a bound of the form $n=\mathcal{O}(kd^2(2\ell+1)^{3w+1})$. This implies in particular that $n=\mathcal{O}(kd^{\mathcal{O}(1)})$ for graphs of constant treewidth and $n=\mathcal{O}(f(k)d^2)$ for chordal graphs, where $f$ is a doubly-exponential function. Using the notion of distance-VC dimension (introduced in 2014 by Bousquet and Thomassé) as a tool, we prove the bounds $n\leq (dk+1)^{t-1}+1$ for $K_t$-minor-free graphs, and $n\leq (dk+1)^{d(3\cdot 2^{r}+2)}+1$ for graphs of rankwidth at most $r$.
For $n \in \mathbb{N}$ let $δ_n$ be the smallest value such that every graph of order $n$ and minimum degree at least $δ_n$ admits an orientation of diameter two. We show that $δ_n=\frac{n}{2} + Θ(\ln n)$.
Let $G$ be a connected graph. The Steiner distance $d(S)$ of a set $S$ of vertices is the minimum size of a connected subgraph of $G$ containing all vertices of $S$. For $k\in \mathbb{N}$, the Steiner $k$-Wiener index $SW_k(G)$ is defined as $\sum_S d(S)$, where the sum is over all $k$-element subsets of the vertex set of $G$. The average Steiner $k$-distance $μ_k(G)$ of $G$ is defined as $\binom{n}{k}^{-1} SW_k(G)$. In this paper we prove upper bounds on the Steiner Wiener index and the average Steiner distance of graphs with given order $n$ and minimum degree $δ$. Specifically we show that $SW_k(G) \leq \frac{k-1}{k+1}\frac{3n}{δ+1} \binom{n}{k} + O(n^{k})$, and that $μ_k(G) \leq \frac{k-1}{k+1}\frac{3n}{δ+1} + O(1)$. We improve this bound for triangle-free graphs to $SW_k(G) \leq \frac{k-1}{k+1}\frac{2n}δ \binom{n}{k} + O(n^{k})$, and $μ_k(G) \leq \frac{k-1}{k+1}\frac{2n}δ + O(1)$. All bounds are best possible.
Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in $D$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We investigate how much one can extend this polynomiality result. We prove that already for digraphs of directed path-width (directed tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand, we show that for digraphs of restricted directed tree-width (directed path-width, DAG-width, respectively) and a fixed integer $k$, the problem of checking whether there is an out-branching with at most $k$ leaves is polynomial time solvable.