It is proved that the minimum-density LDS problem in infinite $\mathbb{Z}$-periodic graphs with a finite period is NP-hard, which bridges the gap between cardinality minimization on finite graphs and density minimization on infinite graphs via a rigorous periodic reduction.
Abstract
A dominating set $S$ of a graph $G$ is a locating-dominating set (LDS) if, for each pair of distinct vertices not in~$S$, their neighbourhoods in $S$ are distinct. Finding a minimum-cardinality LDS in finite graphs is a well-known NP-hard problem. On infinite graphs, this problem naturally generalises to finding an LDS of minimum density. While density bounds have been widely studied for specific infinite regular grids, no computational complexity results exist for infinite graphs. We prove that the minimum-density LDS problem in infinite $\mathbb{Z}$-periodic graphs with a finite period is NP-hard. This result bridges the gap between cardinality minimization on finite graphs and density minimization on infinite graphs via a rigorous periodic reduction. Furthermore, our approach can be adapted to establish NP-hardness for related structural problems on infinite periodic graphs.
For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollob\'as--Erd\H{o}s--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed positive linear degree, and, within every hereditary graph class, a uniform sublinear bound is equivalent to a sublinear bound on graphs of every fixed positive linear vertex connectivity. We prove the sharp general estimate \[ h(G)\le \left\lfloor\frac{|V(G)|}{2\alpha(G)+\delta(G)-|V(G)|}\right\rfloor \] whenever the denominator is positive, with equality for balanced complete multipartite graphs. Consequently, every $3$-colorable graph of order $n$ with $\kappa(G)\ge\rho n$ and $\rho>1/3$ has a hitting set of size at most $\lfloor(\rho-1/3)^{-1}\rfloor$; direct use of a $3$-coloring improves this to $6$ when $\kappa(G)>4n/9$ and to the sharp bound $3$ when $\kappa(G)>n/2$. For dense regular graphs with independence ratio greater than $1/4$, we obtain a logarithmic bound, while constructions with linear degree and linear independence number show that $h(G)=\Omega(\sqrt n)$ can still occur. We also prove a logarithmic bound for near-regular $3$-colorable graphs and exhibit a critical family at connectivity $n/3$ that explains the limitations of the degree-surplus and degree-ratio methods.
A circle graph is the intersection graph of a set of chords in a circle. A dominating set of a graph $G=(V,E)$ is a subset $D\subseteq V$ such that every vertex in $V\setminus D$ is adjacent to at least one vertex of $D$. Computing a minimum dominating set is known to be NP-hard on circle graphs. In this paper, we study the minimum dominating set problem on bipartite circle graphs, namely, circle graphs admitting a chord representation in which the chords can be partitioned into two color classes such that no two chords of the same color intersect. We prove that the problem remains NP-hard for this restricted graph class by a reduction from Planar Monotone 3-SAT. On the positive side, we present a polynomial-time 2-approximation algorithm and develop a polynomial-time approximation scheme (PTAS) based on local search.
A. K. Abu-Affash, Paz Carmi, Joseph S. B. Mitchell· 0 citations
A set of vertices $S$ of a graph $G$ is locating-dominating if $S$ is dominating and, for each pair of distinct vertices not in $S$, their neighborhoods in $S$ are distinct. We present results on the minimum density of such sets in the infinite hexagonal grid with a finite number of rows $k$, also known as the hexagonal strip of width $k$, which we denote by $H_k$. For each $k\geq 2$, we present either an optimal solution or a quasi-optimal solution for $H_k$ that is within $1.3\%$ of the optimum. We describe an exact exponential-time algorithm for fixed k, which we implemented to find optimal solutions for $k \leq 5$. As the infinite grid $H_{k}$ always admits a periodic optimal solution, to deal with larger values of $k$, we present an integer linear program that finds an optimal periodic solution for $H_{k}$ for each fixed period. This program yields high-quality feasible solutions for $H_7$ and $H_8$, which we then combine with an optimal solution for $H_3$ to obtain quasi-optimal solutions for all $k\geq 6$. All these solutions admit a very short description.
In 1962, Erd\H{o}s characterized the maximum size of nonhamiltonian graphs of order $n$ with minimum degree at least $k$. Later, Ning and Peng [Combin. Probab. Comput. 29 (2020) 128-136] extended Erd\H{o}s's results to the clique condition and provided the maximum clique number for nonhamiltonian graphs of order $n$ with minimum degree at least $k$. Recently, Zhang [European J. Combin. 112 (2023) 103728] determined the maximum number of $s$-cliques in nonhamiltonian graphs with prescribed order and minimum degree. A natural extension is to characterize the maximum number of $s$-cliques under other graph properties. Notably, disjoint path cover problems are closely related to Hamiltonicity. In this paper, we generalize results on Hamiltonicity and establish sufficient conditions for a graph to possess one-to-one, one-to-many and many-to-many $t$-disjoint path covers in terms of the number of cliques and the $\alpha$-spectral radius, respectively. Furthermore, we characterize the extremal graphs that attain these bounds respectively.
The spectral Tur\'an type problem, initiated by Nikiforov in 2007, aims to determine the graphs among $n$-vertex $H$-free graphs having maximum spectral radius. In this paper, we study this problem for $1$-planar graphs, i.e., graphs that admit a drawing in the plane such that each edge is crossed at most once. Recently, Xu and Chang proved that the graphs among all $n$-vertex $K_5$-free $1$-planar graphs having maximum spectral radius lie within a small family of candidates. First, this paper explicitly identifies the unique spectral extremal graph among the $n$-vertex $K_5$-free $1$-planar graphs. Second, it establishes a structural reduction theorem: For any forbidden subgraph $F$ with $\delta(F)\ge2$ that is contained in $K_2\vee P_{n-2}^{2+}$ but not in $K_2\vee I_{n-2}$, every spectral extremal $F$-free $1$-planar graph contains a spanning complete bipartite graph $K_{2,n-2}$, where $P^{2+}_{n-2}$ is obtained from a path $u_1u_2\dots u_{n-2}$ by adding edge $u_1u_{n-2}$ and all edges $u_iu_{i+2}$ for $1\le i\le n-4$, and $I_{n-2}$ denotes the empty graph on $n-2$ vertices. As applications, the graph among all $n$-vertex $C_5$-free (resp. $2C_5$-free) $1$-planar graphs having maximum spectral radius is determined. These results extend spectral Tur\'{a}n type problems for $1$-planar graphs from cliques to cycles and their disjoint union.
The neighbourhood complexity $nc(G,k)$ of a graph $G$ is a quantity measuring, for a graph $G$ and an integer $k$, the maximum possible number (over all vertex subsets $S$ of size $k$) $|\{N[v]\cap S, v\in V(G)\}|$ of $S$-neighbourhoods in $G$. This notion is important in structural graph theory and algorithm design (especially in parameterized complexity, in particular model checking and kernelization). While generally $nc(G,k)\leq 2^k$ and this bound can be achieved, it is known that sparse graphs and structured dense graphs have linear neighbourhood complexity, that is, $nc(G,k)\in O(k)$ for any such graph $G$. However, for many graph classes, the best possible constants are not known. We focus on graphs of bounded treewidth and pathwidth, showing that (when $k\geq w+1$) (i) if $G$ has treewidth $w\geq 2$, then $nc(G,k)\leq (k-w+1)2^{w}+w$, and (ii) if $G$ has pathwidth $w\geq 2$, then $nc(G,k)\leq (k-w+2)2^{w-1}+2k-w-2$. Moreover, we provide constructions that reach these bounds, whenever $w\geq 2$ and $k\geq 2w+1$ ($k\geq 2w-1$ for pathwidth). Interestingly, in contrast, we also have the tight bound $nc(G,k)\leq \frac{7}{3}k$, for graphs $G$ with pathwidth 1 or treewidth 1.
Gaétan Berthe, F. Foucaud, Tuomo Lehtilä et al.· 0 citations