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A monitoring edge-geodetic set (or meg-set for short) of a graph is a set of vertices M such that if any edge is removed, then the distance between some two vertices of M increases. This notion was introduced by Foucaud et al. in 2023 as a way to monitor networks for communication failures. As computing a minimum meg-set is hard in general, recent works aimed to find polynomial-time algorithms to compute minimum meg-sets when the input belongs to a restricted class of graphs. Most of these results are based on the property of some classes of graphs to admit a unique minimal meg-set, which is then easy to compute. In this work, we prove that chordal graphs also admit a unique minimal meg-set, answering a standing open question of Foucaud et al.

Arc-Kayles is a game where two players alternate removing two adjacent vertices until no move is left, the winner being the player who played the last move. Introduced in 1978, its computational complexity is still open. More recently, subtraction games, where the players cannot disconnect the graph while removing vertices, were introduced. In particular, Arc-Kayles admits a non-disconnecting variant that is a subtraction game. We study the computational complexity of subtraction games on graphs, proving that they are PSPACE-complete even on very structured graph classes (split, bipartite of any even girth). We give a quadratic kernel for Non-Disconnecting Arc-Kayles when parameterized by the feedback edge number, as well as polynomial-time algorithms for clique trees and a subclass of threshold graphs. We also show that a sufficient condition for a second player-win on Arc-Kayles is equivalent to the graph isomorphism problem.

Given a digraph, an ordering of its vertices defines a backedge graph, namely the undirected graph whose edges correspond to the arcs pointing backwards with respect to the order. The degreewidth of a digraph is the minimum over all ordering of the maximum degree of the backedge graph. We answer an open question by Keeney and Lokshtanov [WG 2024], proving that it is NP-hard to determine whether an oriented graph has degreewidth at most 1, which settles the last open case for oriented graphs. We complement this result with a general discussion on parameters defined using backedge graphs and their relations to classical parameters.

We pursue the study of edge-irregulators of graphs, which were recently introduced in [Fioravantes et al. Parametrised Distance to Local Irregularity. IPEC, 2024]. That is, we are interested in the parameter Ie(G), which, for a given graph G, denotes the smallest k≥0 such that G can be made locally irregular (i.e., with no two adjacent vertices having the same degree) by deleting k edges. We exhibit notable properties of interest of the parameter Ie, in general and for particular classes of graphs, together with parameterized algorithms for several natural graph parameters.

Despite the computational hardness previously exhibited by this problem (NP-hard, W[1]-hard w.r.t. feedback vertex number, W[1]-hard w.r.t. solution size), we present two FPT algorithms, the first w.r.t. the solution size plus Δ and the second w.r.t. the vertex cover number of the input graph.

Finally, we take important steps towards better understanding the behaviour of this problem in dense graphs. This is crucial when considering some of the parameters whose behaviour is still uncharted in regards to this problem (e.g., neighbourhood diversity, distance to clique). In particular, we identify a sub-family of complete graphs for which we are able to provide the exact value of Ie(G). These investigations lead us to propose a conjecture that Ie(G) should always be at most 13m+c, where m is the number of edges of the graph G and c is some constant. This conjecture is verified for various families of graphs, including trees.

Given a c-colored graph G, a vertex v of G is said to be happy if it has the same color as all its neighbors. The notion of happy vertices was introduced by Zhang and Li [27] to compute the homophily of a graph. Eto, Fujimoto, Kiya, Matsushita, Miyano, Murao and Saitoh [11] introduced the Maker-Maker version of the Happy vertex game, where two players compete to claim more happy vertices than their opponent. We introduce here the Maker-Breaker happy vertex game: two players, Maker and Breaker, alternately color the vertices of a graph with their respective colors. Maker aims to maximize the number of happy vertices at the end, while Breaker aims to prevent her. This game is also a scoring version of the Maker-Breaker domination game introduced by Duchene, Gledel, Parreau and Renault [8], as a happy vertex corresponds exactly to a vertex that is not dominated in the domination game. Therefore, this game is a very natural game on graphs and can be studied within the scope of scoring positional games [3]. We initiate here the complexity study of this game, by proving that computing its score is PSPACE-complete on trees, NP-hard on caterpillars, and polynomial on subdivided stars. Finally, we provide the exact value of the score on graphs of maximum degree 2, and we provide an FPT-algorithm to compute the score on graphs of bounded neighborhood diversity. An important contribution of the paper is that, to achieve our hardness results, we introduce a new type of incidence graph called the literal-clause incidence graph for 2-SAT formulas. We prove that QMAX 2-SAT remains PSPACE-complete even if this graph is acyclic, and that MAX 2-SAT remains NP-complete, even if this graph is acyclic and has maximum degree 2, i.e. is a union of paths. We demonstrate the importance of this contribution by proving that Incidence, the scoring positional game played on a graph is also PSPACE-complete when restricted to forests.

Vertex splitting is a graph modification operation in which a vertex is replaced by multiple vertices such that the union of their neighborhoods equals the neighborhood of the original vertex. We introduce and study vertex splitting as a graph modification operation for transforming graphs into interval graphs. Given a graph G and an integer k, we consider the problem of deciding whether G can be transformed into an interval graph using at most k vertex splits. We prove that this problem is NP-hard, even when the input is restricted to subcubic planar bipartite graphs. We further observe that vertex splitting differs fundamentally from vertex and edge deletions as graph modification operations when the objective is to obtain a chordal graph, even for graphs with maximum independent set size at most two. On the positive side, we give a polynomial-time algorithm for transforming, via a minimum number of vertex splits, a given graph into a disjoint union of paths, and that splitting triangle free graphs into unit interval graphs is also solvable in polynomial time.

The classical Maker-Breaker positional game is played on a board which is a hypergraph H, with two players, Maker and Breaker, alternately claiming vertices of H until all the vertices are claimed. When the game ends, Maker wins if she has claimed all the vertices of some edge of H; otherwise, Breaker wins. Playing this game in real life can be done by placing tokens on the vertices of the board. In this paper, we study the unfortunate case in which one or both players do not have enough tokens to cover all the vertices and, as such, will have to move their tokens around at some point instead of placing new ones. There may be a bias, in that Maker and Breaker do not necessarily have the same amount of tokens. The present paper initiates the study of this generalization of positional games, called token positional games. A particularly interesting case is when Maker has a winning strategy in the classical game: what is the lowest number of tokens with which she still wins against Breaker’s unlimited stock? We notably show that, for k-uniform hypergraphs on an arbitrarily large number n of vertices, this number equals k if k ∈ {2, 3} but can vary from k to Ω(n) if k ≥ 4. From an algorithmic point of view, PSPACE-hardness in general is inherited from classical positional games, but we get a polynomial-time algorithm to solve the case where Breaker only has one token. We also establish EXPTIME-completeness for a “token sliding” variation of the game.