[Paper Review] Spectral results for the dominating induced matching problem

Domingos M. Cardoso, Enide Andrade+2 more|arXiv (Cornell University)|Nov 12, 2013

Advanced Graph Theory Research9 references3 citations

TL;DR

This paper characterizes the adjacency, Laplacian, and signless Laplacian spectra of graphs with complete dominating induced matchings (CDIM), establishes polynomial-time recognition of such graphs via their principal eigenvectors, and derives tight upper and lower bounds on the size of dominating induced matchings using eigenvalues. The key contribution is linking spectral graph theory to the algorithmic recognition and structural analysis of CDIM graphs.

ABSTRACT

A matching M is a dominating induced matching of a graph, if every edge of the graph is either in $M$ or has a common end-vertex with exactly one edge in $M$. The concept of complete dominating induced matching is introduced as graphs where the vertex set can be partitioned into two subsets, one of them inducing an $1$-regular graph and the other defining an independent set and such that all the remaining edges connect each vertex of one set to each vertex of the other. The principal eigenvectors of the adjacency, Laplacian and signless Laplacian matrices of graphs with complete dominating induced matchings are characterized and, therefore, the polynomial time recognition of graphs with complete dominating induced matchings is stated. The adjacency, Laplacian and signless Laplacian spectrum of graphs with complete dominating induced matchings are characterized. Finally, several upper and lower bounds on the cardinality of a dominating induced matching obtained from the eigenvalues of the adjacency, Laplacian and signless Laplacian matrices are deduced and examples for which some of these bounds are tight are presented.

Motivation & Objective

• To characterize the spectra of adjacency, Laplacian, and signless Laplacian matrices for graphs with complete dominating induced matchings (CDIM).
• To establish a polynomial-time algorithm for recognizing graphs with CDIM using spectral properties of their principal eigenvectors.
• To derive tight upper and lower bounds on the size of dominating induced matchings based on eigenvalues of graph matrices.
• To explore the structural and spectral implications of CDIM in graphs, particularly through the join operation of induced matchings and independent sets.

Proposed method

• Modeling CDIM graphs as the join of an mK₂ (2m vertices) and a null graph (s vertices), denoted H = H_r ∨ H_s.
• Analyzing the adjacency, Laplacian, and signless Laplacian matrices of such join graphs using block matrix decomposition.
• Using the Rayleigh quotient and trace identities to derive lower bounds on the size of the dominating induced matching m from the largest eigenvalues of A(G), L(G), and Q(G).
• Applying Geršgorin’s theorem to ensure non-negative eigenvalues for L(G) and Q(G), and leveraging symmetry and realness of the matrices.
• Deriving bounds via the trace of L(G) and Q(G), and incorporating minimum degree δ(G) to refine inequalities.
• Validating bounds with explicit examples, including a graph with σ_Q(G) = {1.0000^{[4]}, 1.5858, 2.2679, 4.0000, 4.4142, 5.7321}, yielding m ≥ 2.

Experimental results

Research questions

• RQ1 Can the spectra of adjacency, Laplacian, and signless Laplacian matrices be fully characterized for graphs with complete dominating induced matchings?
• RQ2 Is there a polynomial-time algorithm to recognize graphs with complete dominating induced matchings using spectral properties?
• RQ3 What tight bounds can be derived for the size of a dominating induced matching using the eigenvalues of A(G), L(G), and Q(G)?
• RQ4 How do the principal eigenvectors of A(G), L(G), and Q(G) reflect the structure of graphs with complete dominating induced matchings?
• RQ5 Are there specific graph families for which the derived eigenvalue bounds on |M| are tight?

Key findings

• The adjacency, Laplacian, and signless Laplacian spectra of graphs with complete dominating induced matchings are fully characterized via block matrix structures derived from the join of mK₂ and a null graph.
• Graphs with complete dominating induced matchings can be recognized in polynomial time using the structure of their principal eigenvectors of A(G), L(G), and Q(G).
• A lower bound on |M| is derived from the Laplacian spectrum: m ≥ (tr(L(G)) − n(μ₁ − 2δ(G))) / (2(2δ(G) + 1)).
• A lower bound from the signless Laplacian spectrum is m ≥ (tr(Q(G)) − n(q₁ − 2δ(G))) / (2(2δ(G) − 1)), with a tight example yielding m ≥ 2.
• For the graph in Example 16, the signless Laplacian bound yields m ≥ 1.0685, thus m ≥ 2, demonstrating tightness.
• The paper establishes that the maximum size of a dominating induced matching satisfies |M| ≤ 4 in the example with |Λ⁻| = 4 and |Λ⁺| = 4, confirming the upper bound derived from eigenvalue interlacing.