On star-k-PCGs: exploring class boundaries for small k values

Acta Informatica (2025) 62:17 https://doi.org/10.1007/s00236-025-00485-z RESEARCH On star-k-PCGs: exploring class boundaries for small k values Angelo Monti1 · Blerina Sinaimeri2 Received: 10 February 2024 / Accepted: 5 March 2025 © The Author(s) 2025 Abstract A graph G = (V , E) is a star-k-pairwise compatibility graph (star-k-PCG) if there exists a weight function w : V → R+ and k mutually exclusive  intervals I1 , I2 , . . . Ik , such that there is an edge uv ∈ E if and only if w(u) + w(v) ∈ i Ii . These graphs are related to two important classes of graphs: pairwise compatibility graphs (PCGs) and multithreshold graphs. It is known that for any graph G there exists a k such that G is a star-k-PCG. Thus, for a given graph G it is interesting to know which is the minimum k such that G is a star-kPCG. We define this minimum k as the star number of the graph, denoted by γ (G). Here we investigate the star number of simple graph classes, such as graphs of small size, caterpillars, cycles and grids. Specifically, we determine the exact value of γ (G) for all the graphs with at most 7 vertices. By doing so we show that the smallest graphs with star number 2 are only 4 and have exactly 5 vertices; the smallest graphs with star number 3 are only 3 and have exactly 7 vertices. Next, we provide a construction showing that the star number of caterpillars is one. Moreover, we show that the star number of cycles and two-dimensional grid graphs is 2 and that the star number of 4-dimensional grids is at least 3. Finally, we conclude with numerous open problems. 1 Introduction The categorization of graphs into different classes is fundamental in graph theory and its applications, as it allows for a more structured and focused study of their properties and behaviors. Indeed, each class of graphs, possess unique characteristics that make them suitable for specific problems and applications. In essence, the diversity of graph classes reflects the diversity of real-world problems they are used to model and solve. In this paper, we concentrate on a specific class of graphs, referred to as star-k-pairwise compatibility graphs. A graph G = (V , E) is a star-k-PCG if there exists a weight function w : V → R+ and Angelo Monti and Blerina Sinaimeri contributed equally to this work. A shorter version of this paper appeared as the conference article in [1]. B Blerina Sinaimeri Angelo Monti 1 Computer Science Department, Sapienza University of Rome, Via Salaria 113, 00139 Rome, Italy 2 Department of Business and Management, Luiss University, Viale Romania, 00197 Rome, Italy 0123456789().: V,-vol 123 17 Page 2 of 20 A. Monti, B. Sinaimeri k mutually exclusive  intervals I1 , I2 , . . . Ik , such that there is an edge uv ∈ E if and only if w(u) + w(v) ∈ i Ii . This class serves as a bridge between two well-established graph classes: pairwise compatibility graphs and multithreshold graphs. Connection with pairwise compatibility graphs (PCGs) A graph G is a k-PCG (known also as multi-interval PCG) if and only if there exists a non-negative edge weighted tree T and k mutually exclusive intervals I1 , I2 , . . . , Ik of non-negative reals such that each vertex of G corresponds to a leaf of T and there is an edge between two  vertices in G if and only if the distance between their corresponding leaves in T lies in i Ii (see e.g. [2]). Such tree T is called the k-witness tree of G. The concept of 1-PCGs, also known as PCGs, originated from the problem of reconstructing phylogenetic trees [3]. Moreover, PCGs are a generalization of the well-known k-leaf power graphs [4] and have proven valuable in describing and analyzing evolutionary processes [5]. One of the most important open problems in the field is, whether given an integer k, the k-PCG can be recognized in polynomial time, and it is unknown whether this problem can be solved in polynomial time, even for the case of k = 1. To make progress towards the solution of this problem, restrictions have been made on the topology of the k-witness tree into two main directions: a star and a caterpillar (see e.g. [6–8]). The class of star-k-PCGs is exactly the class of k-PCGs for which the witness tree is a star [1, 9]. Figure 1 depicts an example of a graph that is a star-1-PCG. This topology constraint on the witness tree, has been proven valuable as for star witness trees, the decision problem becomes simpler compared to the general case. Indeed, Xiao and Nagamochi [10] introduced the first polynomial-time algorithm for identifying graphs that are star-1-PCGs. Next, Kobayashi et al. in [11] improved upon this result by introducing a new characterization of star-1-PCGs that led to a linear time recognition algorithm. Connection with multithreshold graphs Multithreshold graphs were introduced by Jamison and Sprague [12] in 2020 as a generalization of the class of threshold graphs introduced by Chvátal and Hammer [13] in 1977 and has since become one of the most prominent and well-studied graph classes (see [14]). In a similar way, multithreshold graphs have gained considerable interest within the research community since their introduction, as evidenced by the following studies [12, 15, 16]. Given real numbers θ1 , θ2 , . . . , θk , with θ1 < θ2 < . . . < θk we say that a graph G = (V , E) is a k-threshold graph with thresholds θ1 , θ2 , . . . , θk if there exist an assignment r : V → R of real ranks to the vertices such that for every pair of distinct vertices u, v ∈ V we have uv ∈ E if and only if the inequality θi ≤ r (u) + r (v) holds for an odd number of indices i. It was shown in [11] that 2-threshold graphs are exactly Fig. 1 a An example of a graph G that is a star-1-PCG graph. In b and c the 1-witness graph and the 1-witness star for which G is a star-1-PCG with I1 = [5, 8] 123 On star-k-PCGs: exploring class boundaries for small k values Page 3 of 20 17 star-1-PCGs. Here we directly extend this claim by showing in Observation 1 that the class of star-k-PCGs is equivalent to the class of 2k-threshold graphs. Results It is already established that every graph G is a star-k-PCG for some positive integer k [2]. Hence, we introduce the following notation. Definition 1 Given a graph G, we define the star number, γ (G), to be the smallest positive integer k, such that G is a star-k-PCG. The star number is not known even for graphs belonging to simple classes. In this paper we firstly focus on n-vertex graphs for small values of n. Identifying the smallest graphs that are excluded from a class not only helps to define the boundaries of that class but could also help towards a characterization of the graph class through forbidden subgraphs. In this framework we consider the following question: what is the smallest value of n for which there exists an n-vertex graph that is not a star-k-PCG? This question has been already investigated for related graphs classes. Indeed, it is known that the smallest graphs that are not 1-PCGs ha (...truncated)