insight - Graph Theory

### Succinct Data Structure for Chordal Graphs with Bounded Vertex Leafage

The authors present a succinct data structure for chordal graphs with bounded vertex leafage that supports efficient adjacency and neighborhood queries.

### Computational Complexity of Recognizing Well-Covered and Generalized Well-Covered Graphs

The core message of this article is to investigate the computational complexity of recognizing well-covered graphs and their generalizations, known as Wk graphs and Es graphs. The authors establish several complexity results, including showing that recognizing Wk graphs and shedding vertices are coNP-complete on well-covered graphs, determining the precise complexity of recognizing 1-extendable (Es) graphs as Θp2-complete, and providing a linear-time algorithm to decide if a chordal graph is 1-extendable.

### Canonical Decompositions of 3-Connected Graphs

Every 3-connected graph can be uniquely decomposed into parts that are either quasi 4-connected, wheels, or thickened K3,m's.

### Tight Lower Bound on the Linear Chromatic Number of Pseudogrids and Improved Relationship Between Centred and Linear Chromatic Numbers

The linear chromatic number of any k × k pseudogrid G is Ω(k), which improves the previously known lower bound and leads to a tighter relationship between the centred chromatic number and the linear chromatic number of graphs.

### Sharp Thresholds for the Existence of Maximum δ-Temporal Cliques in Random Simple Temporal Graphs

The size of a maximum δ-temporal clique in a random simple temporal graph on n vertices is approximately 2 log n / log (1/δ) with high probability.

### Polynomial-Time Algorithms for Hamiltonian Path and Cycle in Graphs with Bounded Independence Number

Hamiltonian path and Hamiltonian cycle problems are solvable in polynomial time for graphs with bounded independence number.

### Efficient Recognition and Analysis of Thick Forests, a Class of Perfect Graphs

Thick forests are a class of perfect graphs that can be recognized in polynomial time, unlike most other classes of thick graphs. The author develops efficient algorithms for recognizing thick forests and analyzing their properties, such as counting independent sets and colorings.

### Efficient Recognition of Chordal and Cograph Cover-Incomparability Graphs

Chordal graphs having at most two independent simplicial vertices are exactly the chordal graphs which are also cover-incomparability (C-I) graphs. A similar result holds for cographs, and linear-time recognition algorithms are developed for these classes of C-I graphs.

### Determining Maximum Values of Sombor-Index-Like Graph Invariants for Trees and Connected Graphs

The paper establishes the maximum values of the Sombor-index-like graph invariants SO5 and SO6 within the set of molecular trees with a given number of vertices, and determines the maximum value of SO5 within the set of graphs obtained by applying the join operation to two specific graphs of a given order.

### Complexity of Dominating Induced Matching and Perfect Edge Domination Problems for Neighborhood Star-Free Graphs

The paper studies the complexity of the dominating induced matching (DIM) problem and the perfect edge domination (PED) problem for neighborhood star-free (NSF) graphs. It proves that the corresponding decision problems are NP-Complete for several subclasses of NSF graphs.