insight - Graph Theory Algorithms

### Efficient Algorithms for λ-Backbone Coloring of Cliques with Tree or Forest Backbones

The paper proposes efficient algorithms to compute the λ-backbone coloring of complete graphs with tree or forest backbones, improving upon previous approximation results.

### Complexity and Algorithms for Matching Cut Problems in Graphs Without Long Induced Paths and Cycles

Matching cut, perfect matching cut, and disconnected perfect matching problems are NP-complete in graphs without induced paths of length 14 or longer, and can be solved in polynomial time in 4-chordal graphs.

### Algorithmic Extensions of Dirac's Theorem on Long Cycles in Graphs with Large Minimum Vertex Degrees

The authors provide an algorithmic generalization of Dirac's theorem, showing that for a 2-connected graph G, deciding whether G contains a cycle of length at least min{2δ(G-B), |V(G)|-|B|} + k can be done in time 2^O(k+|B|) * n^O(1), where B is a subset of vertices and k is an integer.

### Maximizing Vertex-Disjoint and Edge-Disjoint Clique Packings in Bounded Degree Graphs

The authors study the problem of finding a maximum-cardinality set of r-cliques in an undirected graph of fixed maximum degree Δ, subject to the cliques being either vertex disjoint or edge disjoint. They provide a complete complexity classification for both the vertex-disjoint and edge-disjoint variants.

### Deterministic Algorithm for Computing Chromatic Number in 1.9999^n Time under the Asymptotic Rank Conjecture

Under the asymptotic rank conjecture, the chromatic number of an n-vertex graph can be computed deterministically in O(1.99982^n) time.

### Efficient Algorithms for Total Domination and Total Roman Domination in Unit Disk Graphs

This paper proposes efficient approximation algorithms for the Total Dominating Set (TDS) and Total Roman Dominating Set (TRDS) problems in unit disk graphs.

### Lower Bounds for Graph Reconstruction Using Maximal Independent Set Queries

The minimum number of maximal independent set queries required to reconstruct the edges of a hidden graph with n vertices and maximum degree Δ is Ω(Δ^2 log(n/Δ)) for randomized non-adaptive algorithms and Ω(Δ^3 log n / log Δ) for deterministic non-adaptive algorithms.

### Efficient Algorithms for Vizing's Theorem on Bounded Degree Graphs

The paper presents fast sequential and distributed algorithms for finding a proper (Δ+1)-edge-coloring of a graph with maximum degree Δ, with a focus on the case when Δ is constant.

### Efficient Algorithms for Determining Elimination Distance to Bounded Degree on Planar Graphs

The elimination distance of a graph to the class of bounded degree graphs can be efficiently computed on planar graphs.

### Efficient Algebraic Algorithms for Solving the Longest Path Problem on Specific Graph Classes

This paper presents a novel algebraic approach to efficiently solving the Longest Path Problem (LPP) on specific classes of graphs, including trees, uniform block graphs, block graphs, and directed acyclic graphs (DAGs). The authors introduce algebraic conditions and operations that can identify and approximate the solution in polynomial time, without relying on weight or distance functions or being constrained to undirected graphs.