The Bounded-Di\-am\-e\-ter Minimum Spanning Tree Problem (BDMSTP) is defined as follows: Given a graph $G$ where each edge $xy$ has a positive cost $w_{xy}$, the goal is to find an optimal spanning tree of $G$ whose diameter does not exceed a prescribed positive integer $D\geq 2$. Applications of the BDMSTP appear in telecommunication network and fiber optic projects, data compression problems, and distributed systems design. This work revisits search methods for the BDMSTP and proposes alternative combinations of local search moves, intending to select efficient sets of moves and improve the quality of the solutions found in the literature. For small 50- and 100-node instances from the widely-used OR-Library, we obtain exact solutions whose optimal values were so far unknown. Having solved these easier cases, we can then concentrate our efforts on solving more challenging OR-Library graph instances with 250, 500, and 1000 nodes. The process of selecting efficient sets of search moves is done by encapsulating different local search versions (based on the Variable Neighborhood Descent method) in an ILS approach. Some new neighborhoods are also proposed. Computational tests show that these neighborhoods allowed the proposed ILS approach to obtain better results than those presented in the literature.

In this work, we consider combinatorial games in which two players alternately choose vertices from a finite graph until a winning condition is achieved. Specifically, we focus our investigation on the well-known game Kayles, in which the selected vertices must form an independent set, and the player who makes the last valid move wins (i.e., the player who chooses a vertex that completes a maximal independent set). Zermelo's Theorem guarantees that, in this scenario, one of the players has a winning strategy---that is, a sequence of moves that ensures a win regardless of the opponent's choices. Given a graph, the typical decision problem associated with this type of game consists of determining which player has a winning strategy. Answering this question means solving the game. We first consider Kayles played on caterpillars. Since caterpillars are interval graphs, an O(n3)-time algorithm for solving Kayles on this graph class is already known [Bodlaender and Kratsch, 2002]. However, we investigate scenarios in which this time complexity can be reduced to O(1) by extending the periodicity property presented in [Guignard and Sopena, 2009] to caterpillars. We prove that the nimber of any caterpillar is equal to the nimber of an equivalent reduced caterpillar, obtained by appropriately removing certain leaves from the original graph. By partitioning these reduced caterpillars into classes, we show that a period of 34 emerges in each investigated class, allowing the computation of nimbers to scale to graphs with a large number of vertices. We present a sufficient condition for a class of caterpillars to exhibit periodicity 34, using it to identify many periodic classes and to calculate the nimber of their caterpillars in O(1) time. Furthermore, we present an O(n2)-time dynamic programming algorithm for solving Kayles on powers of paths, improving upon the O(n4)-time complexity given in [Bodlaender and Kratsch, 2002]for graphs with an asteroidal number of at most 2. Finally, we show how this same O(n2)-time algorithm can be adapted to solve Kayles on powers of cycles, thereby reducing the O(n3)-time complexity previously established in [Bodlaender and Kratsch, 2002] for circular-arc graphs.

We study the P_3-convexity, the path convexity generated by all three-vertex paths, and focus on the problem of counting the P_3-convex vertex sets of a graph G, denoted by (G). First, we settle the associated extremal question: we characterize the n-vertex graphs maximizing (G) among all graphs and determine the connected extremal graphs. Next, we investigate computational complexity and show that counting P_3-convex sets is #𝖯-complete already on split graphs, even under additional structural restrictions. On the positive side, we identify two tractable subclasses, namely trees and threshold graphs, and obtain linear-time algorithms for both. Finally, we design nontrivial exact exponential-time algorithms for general graphs, combining structural decomposition, propagation rules capturing forced consequences of P_3-convexity, and fast counting of independent sets in auxiliary graphs. The resulting strategy becomes particularly effective on graph classes where large independent sets are guaranteed and can be found efficiently.

The Yutsis property of a graph G is the property of partitioning its vertex set into two induced trees. Although recognizing Yutsis graphs is NP-complete even on planar graphs, it is still possible to consider two even more challenging problems: (i) recognizing k- Yutsis graphs, which are graphs that have their vertex sets partitioned into k induced trees, for a fixed k >= 2; (ii) determining the tree cover number of a given graph G, i.e., the minimum number of vertex-disjoint induced trees covering all vertices of G. We prove that determining the tree cover number of a split graph G is NP-hard, contrasting with the polynomial-time recognition of k-Yutsis chordal graphs. We also investigate the tree cover number computation and the k-Yutsis graph recognition concerning treewidth and clique- width parameterizations. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Graph convexity has been used as an important tool to better understand the structure of classes of graphs. Many studies are devoted to determine if a graph equipped with a convexity is a convex geometry . In this work we survey results on characterizations of well-known classes of graphs via convex geometries. We also give some contributions to this subject.

In this paper, we study two graph convexity parameters: iteration time and general position number. The iteration time was defined in 1981 in the geodesic convexity, but its computational complexity was so far open. The general position number was defined in the geodesic convexity and proved NP-hard in 2018. We extend these parameters to any graph convexity and prove that the iteration number is NP-hard in the P3 convexity. We use this result to prove that the iteration time is also NP-hard in the geodesic convexity even in graphs with diameter two, a long standing open question. These results are also important since they are the last two missing NP-hardness results regarding the ten most studied graph convexity parameters in the geodesic and P3 convexities. We also prove that the general position number of the monophonic convexity is W[1]-hard (parameterized by the size of the solution) and n1−ε-inapproximable in polynomial time for any ε>0 unless P=NP, even in graphs with diameter two. Finally, we also obtain FPT results on the general position number in the P3 convexity and we prove that it is W[1]-hard (parameterized by the size of the solution).

A walk $u_0u_1 \ldots u_{k-1}u_k$ is a \textit{weakly toll walk} if $u_0u_i \in E(G)$ implies $u_i = u_1$ and $u_ju_k\in E(G)$ implies $u_j=u_{k-1}$. A set $S$ of vertices of $G$ is {\it weakly toll convex} if for any two non-adjacent vertices $x,y \in S$ any vertex in a weakly toll walk between $x$ and $y$ is also in $S$. The {\em weakly toll convexity} is the graph convexity space defined over weakly toll convex sets. Many studies are devoted to determine if a graph equipped with a convexity space is a {\em convex geometry}. An \emph{extreme vertex} is an element $x$ of a convex set $S$ such that the set $S\backslash\{x\}$ is also convex. A graph convexity space is said to be a convex geometry if it satisfies the Minkowski-Krein-Milman property, which states that every convex set is the convex hull of its extreme vertices. It is known that chordal, Ptolemaic, weakly polarizable, and interval graphs can be characterized as convex geometries with respect to the monophonic, geodesic, $m^3$, and toll convexities, respectively. Other important classes of graphs can also be characterized in this way. In this paper, we prove that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph. Furthermore, some well-known graph invariants are studied with respect to the weakly toll convexity.

The Yutsis property of a simple, connected, and undirected graph is the property of partitioning its vertex set into two induced trees. Although the first impression is that such a property is quite particular, it is more general than Hamiltonicity on planar graphs since a planar graph satisfies the Yutsis property if and only if its dual is Hamiltonian. Despite the fact that recognizing Yutsis graphs is NP-complete even on planar graphs, it is still possible to consider two even more challenging problems: (i) the recognition of k-Yutsis graphs, which are graphs that have their vertex sets partitioned into k induced trees, for a fixed k≥ 2 ; (ii) to find the minimum number of vertex-disjoint induced trees that cover all vertices of a graph G, which is called the tree cover number of G. The studies on Yutsis graphs emerge from the quantum theory of angular momenta since it appears as a graphical representation of general recoupling coefficients, and the studies on the tree cover number are motivated by its equality with the maximum positive semidefinite nullity on multigraphs with treewidth at most two. Given the interest in the tree cover number on graphs with bounded treewidth, we investigate the parameterized complexity of the tree cover number computation. We prove that the tree cover number can be determined in 2^𝒪(twlog tw)· n^𝒪(1) , where tw is the treewidth of the input graph, but it cannot be solved in 2^o(twlog tw)· n^𝒪(1) time unless ETH fails. Similarly, we conclude that recognizing k-Yutsis graphs can be done in k^𝒪(tw)· n^𝒪(1) time, but it cannot be done in (k-ϵ )^tw· n^𝒪(1) time assuming SETH. We also show that the problem of determining the tree cover number of a graph G is polynomial-time solvable on graphs with bounded clique-width, but it is W[1]-hard considering clique-width parameterization while recognizing k-Yutsis graphs can be done in FPT time. Furthermore, contrasting with the polynomial-time recognition of k-Yutsis chordal graphs, for split graphs G having a partition V(G)=(S,K) where S is an independent set and K is a clique, we prove that determining the tree cover number of G is NP-hard even when S has only vertices of degree 2 or 4, but it is polynomial-time solvable when each vertex of S has either odd degree or degree two in G. We also provide some characterizations for chordal k-Yutsis subclasses.

The influence of individuals in a network and its propagation is dealt with in several studies in the literature. A well-known model is the majority target set, in which if most of the neighbors of an individual in the network are influenced, then the individual is also influenced. Finding a majority target set of minimum size is an NP-hard problem for general graphs. This paper proposes a heuristic for this problem, which has faster runtimes and achieves better solution values than related works, both on small random instances and on large real social network graphs.

In this paper, we investigate two graph convexity parameters: the iteration time and the general position number. Harary and Nieminem introduced in 1981 the iteration time in the geodesic convexity, but its computational complexity was still open. Manuel and Klavˇzar introduced in 2018 the general position number of the geodesic convexity and proved that it is NP -hard to compute. In this paper, we extend these parameters to the P 3 convexity and prove that it is NP -hard to compute them. With this, we also prove that the iteration number is NP -hard on the geodesic convexity even in graphs with diameter two. These results are the last three missing NP -hardness results regarding the ten most studied graph convexity parameters in the geodesic and P 3 convexities.