Related papers: Branching points in the planar Gilbert--Steiner pr…
Let $G$ be a 3-connected planar graph. Define the co-tree of a spanning tree $T$ of $G$ as the graph induced by the dual edges of $E(G)-E(T)$. The well-known cut-cycle duality implies that the co-tree is itself a tree. Let a $k$-tree be a…
Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We…
This thesis is devoted to the study of well-posedness properties of some geometric variational problems: existence, regularity and uniqueness of solutions. We study two specific problems arising in the context of geometric calculus of…
The \emph{Steiner tree} problem is one of the fundamental and classical problems in combinatorial optimization. In this paper, we study this problem in the $\mathcal{CONGESTED}$ $\mathcal{CLIQUE}$ model of distributed computing and present…
We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper…
Let $T$ be a distinguished subset of vertices in a graph $G$. A $T$-\emph{Steiner tree} is a subgraph of $G$ that is a tree and that spans $T$. Kriesell conjectured that $G$ contains $k$ pairwise edge-disjoint $T$-Steiner trees provided…
The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number…
The Steiner distance of vertices in a set $S$ is the minimum size of a connected subgraph that contain these vertices. The sum of the Steiner distances over all sets $S$ of cardinality $k$ is called the Steiner $k$-Wiener index and studied…
Let $R$ and $B$ be two disjoint sets of points in the plane such that $|B|\leqslant |R|$, and no three points of $R\cup B$ are collinear. We show that the geometric complete bipartite graph $K(R,B)$ contains a non-crossing spanning tree…
The Steiner Forest problem is an important generalization of the Steiner Tree problem. We are given an undirected graph with nonnegative edge costs and a collection of pairs of vertices. The task is to compute a cheapest forest with the…
Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded…
We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds…
Various questions related to distances between vertices of simple, finite graphs are of interest to extremal graph theorists. The Steiner distance of a set of $k$ vertices is a natural generalization of the regular distance. We extend…
We present a new exact algorithm for the Steiner tree problem in edge-weighted graphs. Our algorithm improves the classical dynamic programming approach by Dreyfus and Wagner. We achieve a significantly better practical performance via…
Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not…
Branch-and-bound is a typical way to solve combinatorial optimization problems. This paper proposes a graph pointer network model for learning the variable selection policy in the branch-and-bound. We extract the graph features, global…
The Wiener index is maximized over the set of trees with the given vertex weight and degree sequences. This model covers the traditional "unweighed" Wiener index, the terminal Wiener index, and the vertex distance index. It is shown that…
The prize-collecting Steiner tree problem PCSTP is a well-known generalization of the classical Steiner tree problem in graphs, with a large number of practical applications. It attracted particular interest during the latest (11th) DIMACS…
We generalize the joints problem to sets of varieties and prove almost sharp bound on the number of joints. As a special case, given a set of $N$ $2$-planes in $\mathbb{R}^6$, the number of points at which three $2$-planes intersect and…
The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of…