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Monophonic position sets of Cartesian and lexicographic products of graphs

Ullas Chandran S. V., Sandi Klavžar, Neethu P. K., James Tuite

TL;DR

This work investigates the monophonic position number in Cartesian and lexicographic product graphs. It develops a structural framework showing that mp-sets in Cartesian products are restricted to three canonical forms (layered, varied, cliquey) and proves sharp general bounds, with exactness when neither factor contains simplicial vertices. For lexicographic products, it provides a compact formula mp(G ∘ H) expressed in terms of the clique number and the mp-structure of the second factor, yielding precise results in notable cases such as triangle-free factors and complete graphs. The results highlight key differences between general and monophonic position in product graphs and provide practical tools for computing mp in graph constructions.

Abstract

The general position problem in graph theory asks for the number of vertices in a largest set $S$ of vertices of a graph $G$ such that no shortest path of $G$ contains more than two vertices of $S$. The analogous monophonic position problem is obtained from the general position problem by replacing ``shortest path'' by ``induced path.'' In this paper the monophonic position number is studied on Cartesian and lexicographic products of graphs. It is proved that in Cartesian products, a monophonic position set can only be in one of three canonical forms, named layered, varied, and cliquey. The monophonic position number of an arbitrary Cartesian product is bounded from below and above. The two bounds coincide if neither of the factors has simplicial vertices. A formula for the monophonic position number of a lexicographic product is given which only contains the clique number and the structure of monophonic sets of the second factor.

Monophonic position sets of Cartesian and lexicographic products of graphs

TL;DR

This work investigates the monophonic position number in Cartesian and lexicographic product graphs. It develops a structural framework showing that mp-sets in Cartesian products are restricted to three canonical forms (layered, varied, cliquey) and proves sharp general bounds, with exactness when neither factor contains simplicial vertices. For lexicographic products, it provides a compact formula mp(G ∘ H) expressed in terms of the clique number and the mp-structure of the second factor, yielding precise results in notable cases such as triangle-free factors and complete graphs. The results highlight key differences between general and monophonic position in product graphs and provide practical tools for computing mp in graph constructions.

Abstract

The general position problem in graph theory asks for the number of vertices in a largest set of vertices of a graph such that no shortest path of contains more than two vertices of . The analogous monophonic position problem is obtained from the general position problem by replacing ``shortest path'' by ``induced path.'' In this paper the monophonic position number is studied on Cartesian and lexicographic products of graphs. It is proved that in Cartesian products, a monophonic position set can only be in one of three canonical forms, named layered, varied, and cliquey. The monophonic position number of an arbitrary Cartesian product is bounded from below and above. The two bounds coincide if neither of the factors has simplicial vertices. A formula for the monophonic position number of a lexicographic product is given which only contains the clique number and the structure of monophonic sets of the second factor.

Paper Structure

This paper contains 8 sections, 23 theorems, 11 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cartesian products
  4. Structure of $\mathop{\mathrm{mp}}\nolimits$-sets
  5. Varied sets
  6. Cliquey and layered sets
  7. Triangle-free graphs
  8. Lexicographic products

Key Result

Lemma 2.1

Thomas-2024b Let $G$ be a connected graph and $M\subseteq V(G)$ be a monophonic position set. Then $G[M]$ is a disjoint union of $k$ cliques $G[M]=\bigcup_{i=1}^{k} W_{i}$. If $k\geq 2$, then for each $i\in [k]$ any two vertices of $W_i$ have a common neighbour in $G-M$.

Figures (3)

  • Figure 1: Layered (left), varied (middle), and cliquey (right) monophonic position sets
  • Figure 2: Sets $S_i^{\prime}$
  • Figure 3: A monophonic position set of $G\circ H$ need not project to a monophonic position set in $H$

Theorems & Definitions (40)

  • Lemma 2.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Definition 3.5
  • Corollary 3.6
  • ...and 30 more