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Bikernels by monochromatic paths

Dennis J. Diaz-Diaz, Isaías F. de-la-Fuente-Jimenez, Teresa I. Hoekstra-Mendoza, Miguel E. Licona-Velázquez, Victoria Terrones-Segura

TL;DR

This work introduces bikernel by monochromatic paths in bicolored digraphs, extending kernel by monochromatic paths to a two-color setting and exploring when such objects exist across paths, cycles, and their products. It develops necessary and sufficient conditions for bikernels in families of bicolored digraphs, provides a unique bikernel characterization via critical versus supercritical vertices, and derives a BK-colorability criterion via star-decompositions and a cyclic-class bijection. It also analyzes directed cycles with chords and Cartesian product constructions, giving precise parity and color constraints that yield or forbid bikernels. The results connect to category theory through the augmentation of double categories, showing that augmented double categories viewed as bicolored digraphs admit bikernels, linking combinatorial and categorical perspectives and suggesting applications in network modeling.

Abstract

In this paper, we introduce the concept of bikernel by monochromatic paths of a bicolored digraph. This concept is strongly motivated by the existing notions of kernels, kernels by monochromatic paths, and double stable augmented categories. We establish sufficient and necessary conditions for several families of bicolored digraphs to have a bikernel by monochromatic paths. Also, we characterize bicolored digraphs without monochromatic cycles that possess a bikernel by monochromatic paths. Similarly, we characterize bicolored digraphs with monochromatic cycles that also have a bikernel by monochromatic paths. Furthermore, we prove sufficient and necessary conditions for some families of digraphs to be $BK$-colorable. This means that a bicoloration of the digraph exists where the resulting bicolored digraph has a bikernel.

Bikernels by monochromatic paths

TL;DR

This work introduces bikernel by monochromatic paths in bicolored digraphs, extending kernel by monochromatic paths to a two-color setting and exploring when such objects exist across paths, cycles, and their products. It develops necessary and sufficient conditions for bikernels in families of bicolored digraphs, provides a unique bikernel characterization via critical versus supercritical vertices, and derives a BK-colorability criterion via star-decompositions and a cyclic-class bijection. It also analyzes directed cycles with chords and Cartesian product constructions, giving precise parity and color constraints that yield or forbid bikernels. The results connect to category theory through the augmentation of double categories, showing that augmented double categories viewed as bicolored digraphs admit bikernels, linking combinatorial and categorical perspectives and suggesting applications in network modeling.

Abstract

In this paper, we introduce the concept of bikernel by monochromatic paths of a bicolored digraph. This concept is strongly motivated by the existing notions of kernels, kernels by monochromatic paths, and double stable augmented categories. We establish sufficient and necessary conditions for several families of bicolored digraphs to have a bikernel by monochromatic paths. Also, we characterize bicolored digraphs without monochromatic cycles that possess a bikernel by monochromatic paths. Similarly, we characterize bicolored digraphs with monochromatic cycles that also have a bikernel by monochromatic paths. Furthermore, we prove sufficient and necessary conditions for some families of digraphs to be -colorable. This means that a bicoloration of the digraph exists where the resulting bicolored digraph has a bikernel.

Paper Structure

This paper contains 5 sections, 20 theorems, 6 equations, 6 figures.

Table of Contents

  1. Main Results
  2. Paths, cycles and products
  3. Characterization of digraphs with bikernel
  4. Directed cycles with chords
  5. Relationship with Category theory

Key Result

Lemma 1

Let $G$ be a digraph whose arcs are colored with two colors: color 1 and color 2. If $G$ has a bikernel $B\subseteq V(G)$, then the following are sufficient conditions for a vertex $v$ to be an element of the bikernel $B$.

Figures (6)

  • Figure 1: A digraph with bikernel the black vertices.
  • Figure 2: The heart as a bikernel of the blood circulatory system.
  • Figure 3: A bicolored digraph $D$ with a kernel by monochromatic paths but without a bikernel.
  • Figure 4: The graph $C_4\square C_4$ where the gray vertices form a bikernel.
  • Figure 5: A decomposition in stars
  • ...and 1 more figures

Theorems & Definitions (47)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • ...and 37 more