[Paper Review] The decycling numbers of graphs

Sheng Bau, Lowell W. Beineke|ArXiv.org|Mar 19, 2007

Graph Labeling and Dimension Problems2 references4 citations

TL;DR

This paper provides a comprehensive review of the decycling number—the size of the smallest vertex set whose removal makes a graph acyclic—offering bounds, exact values for specific graphs like hypercubes, grids, and snakes, and structural characterizations. Key contributions include NP-completeness of the problem, tight bounds for n-cubes, and connections to graph connectivity and coding theory.

ABSTRACT

For a graph $G$ and $S\subset V(G)$, if $G - S$ is acyclic, then $S$ is said to be a decycling set of $G$. The size of a smallest decycling set of $G$ is called the decycling number of $G$. The purpose of this paper is a comprehensive review of recent results and several open problems on this graph parameter. Results to be reviewed include recent work on decycling numbers of cubes, grids and snakes. A structural description of graphs with a fixed decycling number based on connectivity is also presented. Graphs with small decycling numbers are characterized.

Motivation & Objective

• To systematically review recent results and open problems on the decycling number of graphs.
• To determine exact and tight bounds for the decycling number in specific graph families such as hypercubes, grids, and snakes.
• To characterize graphs with fixed decycling numbers using connectivity and structural properties.
• To explore algorithmic and computational challenges, particularly the NP-completeness of the problem.
• To establish connections between decycling number and other graph parameters like independence and covering numbers.

Proposed method

• Uses the definition of a decycling set S such that G−S is acyclic, with ∇(G) being the minimum size of such a set.
• Applies the formula ∇(G) ≥ (q−p+1)/(Δ−1) for connected graphs with q edges, p vertices, and maximum degree Δ.
• Employs the outlay function θ(S) = σ(S) − |S| − ε(S) − ω(G−S) + 1 to characterize decycling sets, with θ(S) = q−p+1 for any decycling set.
• Utilizes graph operations such as the Cartesian product G×H and vertex identification H∘fJ to construct graphs with known decycling numbers.
• Applies structural decomposition, including identifying graphs with ∇(G)=k as those obtainable by attaching a k-set to a tree in a (k+1)-connected manner.
• Employs recursive and inductive constructions for graphs with small decycling numbers (e.g., ∇(G)=2 or 3), based on connectivity and tree components.

Experimental results

Research questions

• RQ1 What are the exact decycling numbers for n-dimensional hypercubes (Qn) for n ≥ 9?
• RQ2 Which cubic graphs G of order 2n satisfy ∇(G) = ⌈(n+1)/2⌉?
• RQ3 Which cubic planar graphs satisfy the same bound?
• RQ4 Is there a fast algorithm to compute the decycling number of maximal outerplanar graphs or 2-dimensional trees?
• RQ5 What is the relationship between decycling number and graph connectivity, particularly for graphs with ∇(G)=k?

Key findings

• For n ≤ 8, the decycling numbers of n-cubes are exactly known: ∇(Q1)=0, ∇(Q2)=1, ∇(Q3)=3, ∇(Q4)=6, ∇(Q5)=14, ∇(Q6)=28, ∇(Q7)=56, ∇(Q8)=112.
• For Q9 to Q13, the paper provides improved bounds: ∇(Q9) ∈ [224, 312], ∇(Q10) ∈ [448, 606], ∇(Q11) ∈ [896, 1184], ∇(Q12) ∈ [1792, 2224], ∇(Q13) ∈ [3584, 4680].
• The decycling number of the Petersen graph is exactly 3.
• For complete graphs, ∇(Kp) = p−2; for complete bipartite graphs Kr,s with r ≤ s, ∇(Kr,s) = r−1.
• The decycling number of a snake with cell size 4 is exactly ⌊n/3⌋, and results for other cell sizes are derived via transformation.
• For maximal outerplanar graphs of order n, the decycling number satisfies 1 ≤ ∇(G) ≤ ⌊n/3⌋.