2026· International Workshop on Graph-Theoretic Concepts in Computer Science· pp. 30:1-30:16· 0 citations· 39 references
Computer Science
TL;DR
Here, the most natural parameters such as the budgets d and a of the players, the vertex cover number and treewidth of the input graph, and combinations of these parameters are considered, showing that the encoding of the costs and weights of the edges has a considerable influence on the problem complexity.
The restricted edge-connectivity of a graph is the minimum size of an edge-cut whose removal leaves every component with at least two vertices. In 2024, Zhang et al. showed that the restricted edge-connectivity of any optimal $1$-planar graph belongs to $\{8,10,12\}$. In this paper, we exclude $8$ as a possible value, thereby proving that the restricted edge-connectivity is either $10$ or $12$, and both values are attainable. Furthermore, we show that the restricted edge-connectivity of a 6-connected optimal 1-planar graph equals $10$ if and only if the graph contains an edge whose two endvertices both have degree $6$. As a key ingredient, we characterize the structure of vertex-induced subgraphs on $n$ vertices with $4n-9$ edges in optimal 1-planar graphs, and use this characterization to establish a connection between restricted edge-cuts and vertex-cuts in optimal 1-planar graphs.
Licheng Zhang, Zhangdong Ouyang, Yuanqiu Huang et al.· 0 citations
Abstract.
We study the problem of guaranteeing the connectivity of a given graph by protecting or strengthening edges. Herein, a protected edge is assumed to be robust and will not fail, which features a nonuniform failure model. We introduce the [Formula: see text]-Steiner-Connectivity Preservation problem where we protect a minimum-cost set of edges such that the underlying graph maintains [Formula: see text]-edge-connectivity between given terminal pairs against edge failures, assuming at most [Formula: see text] unprotected edges can fail. We design polynomial-time exact algorithms for the cases where [Formula: see text] and [Formula: see text] are small and approximation algorithms for general values of [Formula: see text] and [Formula: see text]. Additionally, we show that when both [Formula: see text] and [Formula: see text] are part of the input, even deciding whether a given solution is feasible is [Formula: see text]-complete. This hardness also carries over to Flexible Network Design, a research direction that has gained significant attention. In particular, previous work focuses on problem settings where either [Formula: see text] or [Formula: see text] is constant, for which our new hardness result now provides justification.
Felix Hommelsheim, Zhenwei Liu, Nicole Megow et al.· SIAM Journal on Discrete Mat...· 0 citations
In the three-terminal Reachability-Preserving Minimum Edge Cut problem, the input is an undirected edge-weighted graph with terminals \(s_1,s_2,t\). The objective is to delete a minimum-cost set of edges that separates \(t\) from both \(s_1\) and \(s_2\), while preserving connectivity between \(s_1\) and \(s_2\). We give a polynomial-time \(O(\log n)\)-approximation algorithm. The algorithm uses a probabilistic distribution of cut-dominating decomposition trees. A direct transfer of a connected tree solution to the original graph is not valid because a connected tree cluster may induce a disconnected vertex set in the graph. We overcome this obstruction by expanding every rooted tree cluster into the connected components it induces in the original graph. These components form a node-weighted auxiliary graph. A minimum node-weighted path in this auxiliary graph produces a connected feasible source side. The main structural observation is that the total graph-boundary cost of all connected components of a rooted tree cluster is no greater than the capacity of the corresponding tree edge. This permits the auxiliary path to be compared with a tree cut separating an optimal preserved \(s_1\)-\(s_2\) path from \(t\). Combining this comparison with the expected \(O(\log n)\) cut distortion of the decomposition trees proves the approximation guarantee.
A matching in a graph G = (V, E) is a set M ⊆ E, such that no two edges in M share an endvertex. An edge cut in G is a set of edges C ⊆ E, such that we can partition V into two non-empty sets R and B, where C is the set of edges with one endvertex in R and one in B. A matching cut is a set of edges M ⊆ E which is both a matching and an edge cut. In this thesis, we consider the decision problems Matching Cut, its variants Disconnected Perfect Matching and Perfect Matching Cut, as well as its generalisation d-Cut. We give polynomial time algorithms and NP-completeness results for certain graph classes, including H-free graphs, for some graphs H, and graphs of bounded radius and diameter. In particular, we solve a 20-year old open problem by showing the NP-completeness of Matching Cut for graphs of high girth. We also consider the maximisation version Maximum Matching Cut, where we ask for a matching cut of maximum size, that is with the maximum number of edges in the matching cut. For this variant we give a complexity dichotomy for graphs of bounded radius, bounded diameter, H-free graphs and bipartite graphs of bounded radius and diameter. We conclude with a comparison of all variants, which allows to identify interesting open problems.
This paper develops an exact algorithm based on the Branch and Bound approach for solving PIDS on chordal graphs, which involves identifying the smallest group of vertices in a given network that maximizes influence throughout the network.
Y A Bekhti, M. Lalou, Méziane Aïder et al.· Pesquisa Operacional· 0 citations
It is proved that the classical cut property always produces a minimum spanning tree of a connected graph, and may be useful for large weighted networks such as communication networks, wiring connections, and transportation networks.
H. Bhapkar, Rezwan Ul Shaban, S. Mir et al.· Journal of the Nigerian Soci...· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
