Jul 2026· Embedded Systems and Applications· pp. 133:1-133:19· 0 citations· 44 references
Computer Science
Abstract
The Weighted Center} problem takes as its input a graph $G=(V,E)$ together with a profile $\pi$ such that every vertex $v$ is mapped to some nonnegative multiplicative weight $\pi(v)$. Its output must be some vertex $c$ minimizing $\max\{\pi(v)d_G(c,v) : v \in V\}$. The classic Center problem corresponds to the case where $\pi(v) =1$ for every vertex $v$. In the literature, various almost linear-time algorithms have been proposed for the Center problem on some well-structured classes of graphs. By contrast, similarly efficient algorithms for the Weighted Center problem have been scarce. We investigate how the Gromov hyperbolicity, alone or in combination with other metric and geometric properties on graphs, can be used in the design of exact and approximate almost linear-time algorithms for the Weighted Center problem. In particular, we derive almost optimal algorithms for the following well-studied classes of graphs: chordal graphs, distance-hereditary graphs (both in $\mathcal{O}(m)$ time), dually chordal graphs and chordal bipartite graphs (both in $\mathcal{O}(m\log{n})$ time).
For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[ \min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, e...
For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollob\'as--Erd\H{o}s--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed pos...
The spectral Tur\'an type problem, initiated by Nikiforov in 2007, aims to determine the graphs among $n$-vertex $H$-free graphs having maximum spectral radius. In this paper, we study this problem for $1$-planar graphs, i.e., graphs that admit a drawing in the plane such that each edge is crossed at most once. Recentl...
A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completi...
This paper investigates $t-spanners that are bounded by certain graph parameters that are asymptotically worst-case optimal and obtains directed $\kappa$-spanners with $\kappa$ being directed tree-width, directed path-width or DAG-width and shows that also in the directed case, this is asymptotically worst-case optimal...
Kevin Buchin, Carolin Rehs, Torben Scheele· 0 citations
For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp exam...
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
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