[Paper Review] A Note on Connected Dominating Set in Graphs Without Long Paths And Cycles

Eglantine Camby, Oliver Schaudt|arXiv (Cornell University)|Mar 12, 2013

Advanced Graph Theory Research12 references3 citations

TL;DR

This paper investigates the ratio between connected domination number (γc) and domination number (γ) in (Pk, Ck)-free graphs for k ≥ 6. It establishes tight upper bounds: γc ≤ γ + 1 for (P6, C6)-free graphs, γc ≤ 2γ for (P8, C8)-free graphs, and shows the general bound γc/γ < 3 is asymptotically sharp in (P9, C9)-free graphs, with constructions achieving these ratios for arbitrarily large γ.

ABSTRACT

The ratio of the connected domination number, $γ_c$, and the domination number, $γ$, is strictly bounded from above by 3. It was shown by Zverovich that for every connected $(P_5,C_5)$-free graph, $γ_c = γ$. In this paper, we investigate the interdependence of $γ$ and $γ_c$ in the class of $(P_k,C_k)$-free graphs, for $k \ge 6$. We prove that for every connected $(P_6,C_6)$-free graph, $γ_c \le γ+ 1$ holds, and there is a family of $(P_6,C_6)$-free graphs with arbitrarily large values of $γ$ attaining this bound. Moreover, for every connected $(P_8,C_8)$-free graph, $γ_c / γ\le 2$, and there is a family of $(P_7,C_7)$-free graphs with arbitrarily large values of $γ$ attaining this bound. In the class of $(P_9,C_9)$-free graphs, the general bound $γ_c / γ\le 3$ is asymptotically sharp.

Motivation & Objective

• To determine tight upper bounds on the ratio γc/γ in (Pk, Ck)-free graphs for k ≥ 6.
• To investigate whether the general bound γc ≤ 3γ is asymptotically tight in specific graph classes.
• To construct infinite families of (Pk, Ck)-free graphs that achieve the derived bounds with arbitrarily large domination numbers.
• To explore the possibility of a characterization of the largest hereditary graph class where γc ≤ 2γ holds.
• To provide constructive proofs enabling polynomial-time algorithms for converting dominating sets into connected dominating sets within these classes.

Proposed method

• Leveraging structural properties of minimal connected dominating sets in (Pk, Ck)-free graphs, particularly their Pk−2-freeness.
• Using known results on connected dominating sets in P6-free graphs, including the existence of spanning complete bipartite subgraphs or C6 in the induced subgraph on the dominating set.
• Applying a constructive approach: given a minimum dominating set D, building a connected dominating set by adding at most O(1) vertices per component of D, depending on the forbidden subgraph constraints.
• Employing case analysis based on the structure of the dominating set's induced subgraph (e.g., whether it contains a C6 or a complete bipartite subgraph).
• Introducing explicit graph constructions (e.g., Hk and Gk) to demonstrate tightness of the bounds by achieving γc/γ = 2 and lim γc/γ = 3.
• Using induction and neighborhood analysis to ensure connectivity and domination in the constructed sets.

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound on the ratio γc/γ in connected (P6, C6)-free graphs?
• RQ2 Can the bound γc ≤ 2γ be achieved in (P8, C8)-free graphs, and is it tight for (P7, C7)-free graphs?
• RQ3 Is the general upper bound γc/γ < 3 asymptotically sharp in the class of (P9, C9)-free graphs?
• RQ4 What is the largest hereditary graph class closed under induced subgraphs where γc ≤ 2γ holds?
• RQ5 Can the structural results be used to design efficient algorithms for computing connected dominating sets from dominating sets in these classes?

Key findings

• For every connected (P6, C6)-free graph, the connected domination number satisfies γc ≤ γ + 1, and this bound is tight for infinitely many graphs with arbitrarily large γ.
• For every connected (P8, C8)-free graph, the ratio γc/γ is at most 2, and this bound is tight for (P7, C7)-free graphs, as demonstrated by an infinite family of such graphs.
• In the class of (P9, C9)-free graphs, the general bound γc/γ < 3 is asymptotically sharp, with a family of graphs Gk satisfying lim_{k→∞} γc(Gk)/γ(Gk) = 3.
• The construction of Hk yields an infinite family of (P7, C7)-free graphs with γc(Hk)/γ(Hk) = 2, proving the tightness of the 2γ bound.
• The construction of Gk, formed by adding a pendant vertex to each end-vertex of Hk, yields (P9, C9)-free graphs with γ(Gk) = k+1 and γc(Gk) = 3k, confirming the asymptotic ratio of 3.
• The proofs are constructive and imply polynomial-time algorithms for transforming a minimum dominating set into a connected dominating set within the respective graph classes.