Related papers: Block Designs and K-Geodetic Graphs: A Survey

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k…

A $k$-regular graph is called a divisible design graph (DDG for short) if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbors, and two…

A maximal geodesic in a graph is a geodesic (alias shortest path) which is not a subpath of a longer geodesic. The geodesic-transversal problem in a graph $G$ is introduced as the task to find a smallest set $S$ of vertices of $G$ such that…

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 $k$-ranking of a graph $G$ is a labeling of its vertices from $\{1,\ldots,k\}$ such that any nontrivial path whose endpoints have the same label contains a larger label. The least $k$ for which $G$ has a $k$-ranking is the ranking number…

The metric dimension, $\dim(G)$, of a graph $G$ is a graph parameter motivated by robot navigation that has been studied extensively. Let $G$ be a graph with vertex set $V(G)$, and let $d(x,y)$ denote the length of a shortest $x-y$ path in…

For a graph G=(V,E), the k-dominating graph of G, denoted by $D_{k}(G)$, has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of $D_{k}(G)$ are adjacent if and only if the dominating set…

A graph is a cograph if it does not contain a 4-vertex path as an induced subgraph. An $(s, k)$-polar partition of a graph $G$ is a partition $(A, B)$ of its vertex set such that $A$ induces a complete multipartite graph with at most $s$…

A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if, for any sequence of k distinct vertices v_1, ..., v_k of G, there exists a cycle (respectively, a hamiltonian cycle) in G containing these k vertices in the specified…

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First…

In the $k$-Center problem, we are given a graph $G=(V,E)$ with positive edge weights and an integer $k$ and the goal is to select $k$ center vertices $C \subseteq V$ such that the maximum distance from any vertex to the closest center…

Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…

Given a graph $G$, we define ${\bf bcg}(G)$ as the minimum $k$ for which $G$ can be contracted to the uniformly triangulated grid $\Gamma_{k}$. A graph class ${\cal G}$ has the SQG${\bf C}$ property if every graph $G\in{\cal G}$ has…

We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an…

Finding k disjoint paths (kDP) is a fundamental problem in graph analysis. For vertices s and t, paths from s to t are said to be disjoint if any two of them share no common vertex except s and t. In practice, disjoint paths are widely…

Let $G=(V,E)$ be a graph of order $n$ and let $1\leq k< n$ be an integer. The $k$-token graph of $G$ is the graph whose vertices are all the $k$-subsets of $V$, two of which are adjacent whenever their symmetric difference is a pair of…

We consider a the minimum k-way cut problem for unweighted graphs with a size bound s on the number of cut edges allowed. Thus we seek to remove as few edges as possible so as to split a graph into k components, or report that this requires…

A set of vertices $W$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W$. A metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. A bipartite graph G(n,n) is…

For any metric $d$ on $\mathbb{R}^2$, an ($\mathbb{R}^2,d$)-geometric graph is a graph whose vertices are points in $\mathbb{R}^2$, and two vertices are adjacent if and only if their distance is at most 1. If $d=\|.\|_{\infty}$, the metric…

An injective $k$-edge-coloring of a graph $G$ is an assignment of colors, i.e. integers in $\{1, \ldots , k\}$, to the edges of $G$ such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors.…