On Bipartite-Almost Bipartite Graphs and the Determinantal Factorization

Abstract

A graph is almost bipartite if it contains exactly one odd cycle, and it is Konig-Egervary if the sum of the independence number and the matching number equals the order of the graph. We introduce the class of Bipartite-Almost Bipartite graphs (BAB-graphs), defined through a controlled union of a bipartite graph and several almost bipartite non-Konig-Egervary graphs. This family unifies and generalizes the previously studied classes of almost bipartite non-Konig-Egervary and R-disjoint graphs. While an almost bipartite non-Konig-Egervary graph contains a single odd cycle, an R-disjoint graph has exactly k pairwise disjoint odd cycles. A BAB-graph may contain many odd cycles that are not necessarily disjoint. We describe the structure of BAB-graphs by means of the Gallai-Edmonds decomposition and obtain explicit expressions for nucleus(G), diadem(G), and ker(G), which allow us to extend several known results for the previous classes. Moreover, we show that the determinant of the adjacency matrix of a BAB-graph can be factorized in terms of the determinants of the adjacency matrices of its component graphs. As a consequence, we confirm the conjecture stating the validity of this factorization for R-disjoint graphs. Finally, we derive combinatorial consequences of these results and establish new bounds for |corona(G)| + |ker(G)|.

Figures (4)

• Figure 1: In this example, a BAB-graph that is not an $R$-disjoint graph is shown.
• Figure 2: Example of \ref{['x14']}: in the Figure, a BAB-graph $G=(B,G_{1},G_{2})$ is shown, where $\det(G)=\det(B)\det(G_{1})\det(G_{2})=4\cdot8\cdot(-2)=-64.$
• Figure 3: In this example, a BAB-graph $G=(G,G_{1})$ is shown; in red we display a maximum matching of $G$, and in blue the vertices of $\text{corona}(G)$. Notice that $\left|\text{corona}(G)\right|+\left|\text{ker}(G)\right|=8+2=10<11=2\cdot5+1=2\alpha(G)+k$.
• Figure 4: Illustration of the proof of \ref{['16']}.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 3.1: edmonds1965pathsgallai1964maximale Gallai--Edmonds structure theorem
• Lemma 3.2
• Lemma 3.3
• Theorem 3.4
• Lemma 3.5
• Corollary 3.6
• Lemma 3.7
• proof