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About 10,000 papers for “Complexity and Algorithms in Graphs”

Non-Clashing Teaching in Graphs: Algorithms, Complexity, and Bounds

Sujoy Bhore, Liana Khazaliya, Fionn Mc Inerney · ArXiv.org · 2026

Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and proved that it is the most efficient batch machine teaching model satisfying the collusion-avoidance benchmark established in the seminal work of Goldman and Mathias [COLT 1993]. Recently, (positive) non-clashing teaching was thoroughly studied for balls in graphs, yielding numerous algorithmic and combinatorial results. In particular, Chalopin et al. [COLT 2024] and Ganian et al. [ICLR 2025] gave an almost complete picture of the complexity landscape of the positive variant, showing that it is tractable only for restricted graph classes due to the non-trivial nature of the problem and concept class. In this work, we consider (positive) non-clashing teaching for closed neighborhoods in graphs. This concept class is not only extensively studied in various related contexts, but it also exhibits broad generality, as any finite binary concept class can be equivalently represented by a set of closed neighborhoods in a graph. In comparison to the works on balls in graphs, we provide improved algorithmic results, notably including FPT algorithms for more general classes of parameters, and we complement these results by deriving stronger lower bounds. Lastly, we obtain combinatorial upper bounds for wider classes of graphs.

Cited by 0Open access

On maximal Roman domination in graphs: complexity and algorithms

Zehui Shao, Yonghao Song, Qiyun Liu et al. · RAIRO. Operations research · 2024

For a simple undirected connected graph G = ( V, E ), a maximal Roman dominating function (MRDF) of G is a function f : V ( G ) → {0, 1, 2} with the following properties: ( i ) For every vertex v ∈ { v ∈ V | f ( v ) = 0}, there exists a vertex u ∈ N ( v ) such that f ( u ) = 2. ( ii ) The set { v ∈ V|f ( v ) = 0} is not a dominating set of G ; In other words, there exists a vertex v ∈ { v ∈ V | f ( v ) ≠ 0} such that N ( v ) ∩ { u ∈ V | f ( u ) = 0} ∅ . The weight of an MRDF of G is the sum of its function values over all vertices, denoted as f ( G ) = ∑ v ∈ V ( G ) f ( v ), and the maximal Roman domination number of G , denoted by γ mR ( G ), is the minimum weight of an MRDF of G . In this paper, we establish some bounds of the maximal Roman domination number of graphs. Additionally, we develop an integer linear programming formulation to compute the maximal Roman domination number of any graph. Furthermore, we prove that maximal Roman domination problem (MRD) is NP-complete even restricted to star convex bipartite graphs and chordal bipartite graphs. Lastly, we show the maximal Roman domination number of threshold graphs, trees, and block graphs can be computed in linear time.

Cited by 1Open access

Non-Clashing Teaching in Graphs: Algorithms, Complexity, and Bounds

Bhore, Sujoy, Khazaliya, Liana, Mc Inerney, Fionn · Zenodo (CERN European Organization for Nuclear Research) · 2026

Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and proved that it is the most efficient batch machine teaching model satisfying the collusion-avoidance benchmark established in the seminal work of Goldman and Mathias [COLT 1993]. Recently, (positive) non-clashing teaching was thoroughly studied for balls in graphs, yielding numerous algorithmic and combinatorial results. In particular, Chalopin et al. [COLT 2024] and Ganian et al. [ICLR 2025] gave an almost complete picture of the complexity landscape of the positive variant, showing that it is tractable only for restricted graph classes due to the non-trivial nature of the problem and concept class. In this work, we consider (positive) non-clashing teaching for closed neighborhoods in graphs. This concept class is not only extensively studied in various related contexts, but it also exhibits broad generality, as any finite binary concept class can be equivalently represented by a set of closed neighborhoods in a graph. In comparison to the works on balls in graphs, we provide improved algorithmic results, notably including FPT algorithms for more general classes of parameters, and we complement these results by deriving stronger lower bounds. Lastly, we obtain combinatorial upper bounds for wider classes of graphs.

Cited by 0Open access

Complexity and algorithms for injective edge-coloring in graphs

Foucaud, Florent, Hocquard, Hervé, Lajou, Dimitri · arXiv (Cornell University) · 2021

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. The problem of determining whether such a $k$-coloring exists is called k-INJECTIVE EDGE-COLORING. We show that 3-INJECTIVE EDGE-COLORING is NP-complete, even for triangle-free cubic graphs, planar subcubic graphs of arbitrarily large girth, and planar bipartite subcubic graphs of girth~6. 4-INJECTIVE EDGE-COLORING remains NP-complete for cubic graphs. For any $k\geq 45$, we show that k-INJECTIVE EDGE-COLORING remains NP-complete even for graphs of maximum degree at most $5\sqrt{3k}$. In contrast with these negative results, we show that \InjPbName{k} is linear-time solvable on graphs of bounded treewidth. Moreover, we show that all planar bipartite subcubic graphs of girth at least~16 are injectively $3$-edge-colorable. In addition, any graph of maximum degree at most $\sqrt{k/2}$ is injectively $k$-edge-colorable.

Cited by 0Open access

Non-Clashing Teaching in Graphs: Algorithms, Complexity, and Bounds

Bhore, Sujoy, Khazaliya, Liana, Mc Inerney, Fionn · Zenodo (CERN European Organization for Nuclear Research) · 2026

Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and proved that it is the most efficient batch machine teaching model satisfying the collusion-avoidance benchmark established in the seminal work of Goldman and Mathias [COLT 1993]. Recently, (positive) non-clashing teaching was thoroughly studied for balls in graphs, yielding numerous algorithmic and combinatorial results. In particular, Chalopin et al. [COLT 2024] and Ganian et al. [ICLR 2025] gave an almost complete picture of the complexity landscape of the positive variant, showing that it is tractable only for restricted graph classes due to the non-trivial nature of the problem and concept class. In this work, we consider (positive) non-clashing teaching for closed neighborhoods in graphs. This concept class is not only extensively studied in various related contexts, but it also exhibits broad generality, as any finite binary concept class can be equivalently represented by a set of closed neighborhoods in a graph. In comparison to the works on balls in graphs, we provide improved algorithmic results, notably including FPT algorithms for more general classes of parameters, and we complement these results by deriving stronger lower bounds. Lastly, we obtain combinatorial upper bounds for wider classes of graphs.

Cited by 0Open access

On maximal Roman domination in graphs: complexity and algorithms

Zehui Shao, Yonghao Song, Qiyun Liu et al. · Springer Link (Chiba Institute of Technology) · 2024

For a simple undirected connected graph G = (V, E), a maximal Roman dominating function (MRDF) of G is a function f : V (G) → {0, 1, 2} with the following properties: (i) For every vertex v ∈ {v ∈ V|f(v) = 0}, there exists a vertex u ∈ N(v) such that f(u) = 2. (ii) The set {v ∈ V|f(v) = 0} is not a dominating set of G; In other words, there exists a vertex v ∈ {v ∈ V|f(v) ≠ 0} such that N(v) ∩ {u ∈ V|f(u) = 0} ∅. The weight of an MRDF of G is the sum of its function values over all vertices, denoted as f(G) = ∑v∈V (G) f(v), and the maximal Roman domination number of G, denoted by γmR(G), is the minimum weight of an MRDF of G. In this paper, we establish some bounds of the maximal Roman domination number of graphs. Additionally, we develop an integer linear programming formulation to compute the maximal Roman domination number of any graph. Furthermore, we prove that maximal Roman domination problem (MRD) is NP-complete even restricted to star convex bipartite graphs and chordal bipartite graphs. Lastly, we show the maximal Roman domination number of threshold graphs, trees, and block graphs can be computed in linear time.

Cited by 0

Total 2-Rainbow Domination in Graphs: Complexity and Algorithms

Manjay Kumar, P. Venkata Subba Reddy · International Journal of Foundations of Computer Science · 2023

For a simple, undirected graph [Formula: see text] without isolated vertices, a function [Formula: see text] which satisfies the following two conditions is called a total 2-rainbow dominating function (T2RDF) of [Formula: see text]. (i) For all [Formula: see text], if [Formula: see text] then [Formula: see text] and (ii) Every [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. The weight of a T2RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. The total 2-rainbow domination number is the minimum weight of a T2RDF on [Formula: see text] and is denoted by [Formula: see text]. The minimum total 2-rainbow domination problem (MT2RDP) is to find a T2RDF of minimum weight in the input graph. In this article, we show that the problem of deciding if [Formula: see text] has a T2RDF of weight at most [Formula: see text] for star convex bipartite graphs, comb convex bipartite graphs, split graphs and planar graphs is NP-complete. On the positive side, we show that MT2RDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, we show that MT2RDP cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text] and also propose [Formula: see text]-approximation algorithm for it. Further, we show that MT2RDP is APX-complete for graphs with maximum degree 4. Next, it is shown that domination problem and the total 2-rainbow domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MT2RDP is presented.

Cited by 0

Polyhedral combinatorics, complexity and algorithms for k-clubs in graphs

Mahdavi Pajouh, Foad · SHAREOK (University of Oklahoma; Oklahoma State University; Central Oklahoma University) · 2013

A k-club is a distance-based graph-theoretic generalization of clique, originally introduced to model cohesive subgroups in social network analysis. The k-clubs represent low diameter clusters in graphs and are suitable for various graph-based data mining applications. Unlike cliques, the k-club model is nonhereditary, meaning every subset of a k-club is not necessarily a k-club. This imposes significant challenges in developing theory and algorithms for optimization problems associated with k-clubs.

Cited by 0