Complexity of Deciding the Equality of Matching Numbers

TL;DR

The paper investigates when the three core matching parameters satisfy equalities, notably $\nu(G)$, $\nu_d(G)$, and $\nu_s(G)$. It combines NP-hardness reductions (notably from Exact Cover By 3-Sets) with structural graph theory to delineate a rich complexity landscape across diameter-bounded and bounded-degree graphs, including a co-NP-completeness result for $\nu_d(G)=\nu_s(G)$ and a polynomial-time algorithm for certain diameter-3 and bounded-degree cases. A central technical contribution is the degree-bounded reduction that preserves the $\nu=\nu_d$ relation, enabling NP-hardness for subcubic bipartite graphs. The work further provides a complete diameter-3 characterization for $\nu=\nu_d$ recognition, and a Cameron–Walker-based framework yielding a polynomial-time test when $\nu_s$ is large relative to the maximum degree; it also proves a flexible sequence realization result for $\nu_{d,i}(G)$, underscoring the broad expressiveness of disconnected matching numbers. These results collectively advance understanding of restricted matching problems and offer practical recognition and testing tools for special graph classes.

Abstract

A matching is said to be disconnected if the saturated vertices induce a disconnected subgraph and induced if the saturated vertices induce a 1-regular graph. The disconnected and induced matching numbers are defined as the maximum cardinality of such matchings, respectively, and are known to be NP-hard to compute. In this paper, we study the relationship between these two parameters and the matching number. In particular, we discuss the complexity of two decision problems; first: deciding if the matching number and disconnected matching number are equal; second: deciding if the disconnected matching number and induced matching number are equal. We show that given a bipartite graph with diameter four, deciding if the matching number and disconnected matching number are equal is NP-complete; the same holds for bipartite graphs with maximum degree three. We characterize diameter three graphs with equal matching number and disconnected matching number, which yields a polynomial time recognition algorithm. Afterwards, we show that deciding if the induced and disconnected matching numbers are equal is co-NP-complete for bipartite graphs of diameter 3. When the induced matching number is large enough compared to the maximum degree, we characterize graphs where these parameters are equal, which results in a polynomial time algorithm for bounded degree graphs.

Figures (13)

• Figure 1: Reduction graph for the input $X = \{x_1,\ldots,x_6\}$ and $\mathcal{C} = \{C_1,C_2,C_3,C_4\}= \{ \{x_1,x_2,x_3\},\\ \{ x_1,x_2,x_4 \}, \{ x_1,x_2,x_5 \}, \{ x_4,x_5,x_6 \} \}$, including a maximum matching corresponding to the exact cover $\mathcal{C}'=\{ \{x_1,x_2,x_3\}, \{x_4,x_5,x_6\} \}$.
• Figure 2: $H(3,5)$
• Figure 3: Example for $q=3$ and $|\mathcal{C}|=5$: $H\cong K_{3,5}$$\rightarrow$$H(3,5)$
• Figure 4: $\Pi(v,k)$
• Figure 5: $u^+_{j} u_j u^-_{j}$$\rightarrow$$U_{j}$

Theorems & Definitions (47)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6