Maximal and maximum dissociation sets in general and triangle-free graphs

A classic result in Euclidean geometry asserts that every non-collinear set of n points in the Euclidean plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with an appropriate definition of line. This conjecture remains open even for graph metrics. In this paper, sufficient conditions on the girth to guarantee that a graph satisfies the conjecture are stated. We first study the existence of a universal line generated by an edge. Then, we focus on graphs of order n and maximum girth g with respect to their radius r. We prove that graphs with minimum degree δ ≥ 4 and girth g = 2r+1 or g = 2r have at least n distinct lines and we also partially solve the case when δ = 3. Graphs with cut-vertices are also studied, providing several upper bounds on the diameter, in terms of the girth, in order to assure that the graph satisfies the Chen-Chvátal conjecture. Finally, we prove that any triangle-free graph of order n, diameter 3 and minimum degree δ ≥ 4 has more than n distinct lines and we also partially solve the case when δ = 3.

Let G = ( V , E ) be a finite undirected graph with vertex set V ( G ) of order | V ( G )| = n and edge set E ( G ) of size | E ( G )| = m . Let Δ = d 1 ≥ d 2 ≥ ⋯ ≥ d n = δ be the degree sequence of the graph G . A clique in a graph G is a complete subgraph of G . The clique number of a graph G , denoted by ω ( G ) , is the order of a maximum clique of G . In 1907 Mantel proved that a triangle-free graph with n vertices can contain at most ⌊ n 2 /4⌋ edges. In 1941 Tur'an generalized Mantel’s result to graphs not containing cliques of size r by proving that graphs of order n that contain no induced K r have at most (1 − 1/ r − 1) n 2 /2 edges. In this paper, we give new bounds for the maximum number of edges in a K r -free graph G of order n , minimum degree δ , and maximum degree Δ . We show that, for the families of graphs having the above properties, our bounds are slightly better than the more general bounds of Tur'an.

On the generation of bicliques of a graph

A vertex subset I of a graph G is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from I. The k -Path Vertex Cover Reconfiguration (k -PVCR) problem asks if one can transform one k-path vertex cover into another via a sequence of k-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of k -PVCR from the viewpoint of graph classes under the well-known reconfiguration rules: \(\mathsf {TS}\), \(\mathsf {TJ}\), and \(\mathsf {TAR}\). The problem for \(k=2\), known as the Vertex Cover Reconfiguration (VCR) problem, has been well-studied in the literature. We show that certain known hardness results for VCR on different graph classes including planar graphs, bounded bandwidth graphs, chordal graphs, and bipartite graphs, can be extended for k -PVCR. In particular, we prove a complexity dichotomy for k -PVCR on general graphs: on those whose maximum degree is 3 (and even planar), the problem is \(\mathtt {PSPACE}\)-complete, while on those whose maximum degree is 2 (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for k -PVCR on trees under each of \(\mathsf {TJ}\) and \(\mathsf {TAR}\). Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given k-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest.

In graph theory, a “dominating vertex set” is a subset of vertices in a graph such that every vertex in the graph is either a member of the subset or adjacent to a member of the subset. In other words, the vertices in the dominating set “dominate” the remaining vertices in the graph. Dominating vertex sets are important in graph theory because they can help us understand and analyze the behavior of a graph. For example, in network analysis, a set of dominant vertices may represent key nodes in a network that can influence the behavior of other nodes. Identifying dominant sets in a graph can also help in optimization problems, as it can help us find the minimum set of vertices that can control the entire graph. Now that there are theories about vagueness, it is important to define parallel ideas in vague structures, such as intuitionistic fuzzy graphs. This paper describes a better way to find dominating vertex sets (DVSs) in intuitive fuzzy graphs (IFGs). Even though there is already an algorithm for finding DVSs in IFGs, it has some problems. For example, it does not take into account the vertex volume, which has a direct effect on how DVSs are calculated. To address these limitations, we propose a new algorithm that can handle large-scale IFGs more efficiently. We show how effective and scalable the method is by comparing it to other methods and applying it to water flow. This work’s contributions can be used in many areas, such as social network analysis, transportation planning, and telecommunications.

As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a specific infinite sequence of graphs or contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that, if a (not necessarily maximal) triangle-free graph of order n has minimum degree δ[ges ]n/3, then the graph is either homomorphic with a member of the indicated family or contains the Petersen graph with one edge contracted. As a corollary we get a recent result due to Chen, Jin and Koh. Finally, we show that every triangle-free graph with δ>n/3 is either homomorphic with C5 or contains the Möbius ladder. A major tool is the observation that every triangle-free graph with δ[ges ]n/3 has a unique maximal triangle-free supergraph.

A graph G is called triangle-free if G does not contain a triangle as an induced subgraph. Let H n be the set of triangle-free graphs of order n with six non-zero eigenvalues. In this paper, we find 19 graphs of H n , and we show that the other graphs of H n can be constructed from these 19 graphs by adding some congruent vertices. Hence we completely characterize the triangle-free graphs with six non-zero eigenvalues.

On 4-[formula omitted]-critical graphs of diameter two

Eigenvalue multiplicity in triangle-free graphs

The maximum number of complete subgraphs in a graph with given maximum degree

The number of maximal independent sets in connected triangle-free graphs

formula omitted]-domination in graphs

We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter tractable algorithms for Dominating Set, t -Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W[i]-hard for some i≥1 in general graphs. We also show that the Dominating Set problem is W[2]-hard for bipartite graphs and hence for triangle free graphs. In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, parameterized by k, is W[1]-hard even on graphs with girth at least six. Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph.

On graphs with the maximum edge metric dimension

Finding maximum independent sets in graphs with bounded maximum degree is a well-studied NP-complete problem. We study two approaches for finding approximate solutions, and obtain several improved performance ratios.The first is a subgraph removal schema introduced in our previous paper. Using better component algorithms, we obtain an efficient method with a Δ/6(1+o(1)) performance ratio. We then produce an implementation of a theorem of Ajtai et al. on the independence number of clique-free graphs, and use it to obtain a O(Δ/loglogΔ) performance ratio with our schema. This is the first o(Δ) ratio.The second is a local search method of Berman and Fürer for which they proved a fine performance ratio but by using extreme amounts of time. We show how to substantially decrease the computing requirements while maintaining the same performance ratios of roughly (Δ+3)/5 for graphs with maximum degree Δ. We then show that a scaled-down version of their algorithm yields a (Δ+3)/4 performance, improving on previous bounds for reasonably efficient methods.KeywordsLocal SearchMaximum DegreePerformance RatioFree GraphIndependence NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.