Generalized Edmonds-Sterboul-Deming configurations Part 3: Determinantal multiplicativity of the SD-KE decomposition of matchable graphs

Abstract

In this work it is shown that the SD-KE decomposition is multiplicative under determinantal-type functions for graphs with perfect matchings, providing a new tool for the study of unimodular and singular matchable graphs.

Figures (6)

• Figure 1: $M$-Jposy
• Figure 2: The blue matching is $M_{1}$ and the red one is $M_{2}$. Note that $R(M_1,1) = R(M_2,1) = \{2,4,5,7\}$ and $R(M_1,2) = R(M_2,2) = \{1,3,6,8\}$
• Figure 3: The red matching is $M_{1}$ and the blue one is $M_{2}$. Note that $R(M_1,1) = R(M_2,1) = \{1,2,3,4,5,6,7,8\}$ and $R(M_1,2) = R(M_2,2) = \{1,2,3,4,5,6,7,8\}$
• Figure 4: Stability under edge deletion.
• Figure 5: Cases of proof of \ref{['Main_result']}

Theorems & Definitions (18)

• Lemma 2.1: kevin2025posy
• Theorem 2.2: harary1962determinant
• Lemma 3.1: Reachability Lemma
• Lemma 3.2: kevin2025posy
• Corollary 3.3
• Theorem 3.4
• proof
• Corollary 3.5
• Theorem 3.6
• proof