[Paper Review] Total Forcing Sets in Trees

Randy Davila, Michael A. Henning|arXiv (Cornell University)|Feb 21, 2017

Advanced Graph Theory Research25 references3 citations

TL;DR

This paper establishes tight upper and lower bounds on the total forcing number $ F_t(T) $ of trees $ T $, proving that $ F_t(T) \leq \frac{1}{\Delta}((\Delta-1)n + 1) $ with equality for a specific family of trees, and $ F_t(T) \geq n_1 $ with equality when $ T $ is a path or a star-like tree. It further shows $ F_t(T) \geq F(T) + 1 $, with equality if and only if $ T $ belongs to a defined family $ \mathcal{H} $, and constructs trees where the gap between $ F_t(T) $ and $ F(T) $ can be arbitrarily large.

ABSTRACT

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. If the initial set $S$ has the added property that it induces a subgraph of $G$ without isolated vertices, then $S$ is called a total forcing set in $G$. The minimum cardinality of a total forcing set in $G$ is its total forcing number, denoted $F_t(G)$. We prove that if $T$ is a tree of order $n \ge 3$ with maximum degree~$Δ$, then $F_t(T) \le \frac{1}Δ((Δ- 1)n + 1)$, and we characterize the infinite family of trees achieving equality in this bound. We also prove that if $T$ is a non-trivial tree with $n_1$ leaves, then $F_t(T) \ge n_1$, and we characterize the infinite family of trees achieving equality in this bound. As a consequence of this result, the total forcing number of a non-trivial tree is strictly greater than its forcing number. In particular, we prove that if $T$ is a non-trivial tree, then $F_t(T) \ge F(T)+1$, and we characterize extremal trees achieving this bound.

Motivation & Objective

• To determine the maximum possible total forcing number $ F_t(T) $ for trees of order $ n $ and maximum degree $ \Delta $.
• To characterize the infinite family of trees achieving the upper bound on $ F_t(T) $ in terms of $ n $ and $ \Delta $.
• To establish a lower bound $ F_t(T) \geq n_1 $, where $ n_1 $ is the number of leaves, and characterize trees achieving equality.
• To investigate the relationship between the total forcing number $ F_t(T) $ and the standard forcing number $ F(T) $, proving $ F_t(T) \geq F(T) + 1 $ with equality iff $ T \in \mathcal{H} $.
• To construct trees where the difference $ F_t(T) - F(T) $ can be made arbitrarily large, demonstrating the gap between the two invariants.

Proposed method

• Derive an upper bound on $ F_t(T) $ using structural decomposition of trees and degree-based analysis, showing $ F_t(T) \leq \frac{1}{\Delta}((\Delta-1)n + 1) $.
• Characterize extremal trees achieving equality in the upper bound by analyzing vertex degrees and component structures, particularly focusing on vertices of maximum degree.
• Prove the lower bound $ F_t(T) \geq n_1 $ using induction and structural properties of trees, showing that every leaf must be forced by a neighbor in any total forcing set.
• Use the concept of trimmed trees $ \mathrm{trim}(T) $ to reduce the problem to trees without leaves, preserving both $ F_t(T) $ and $ F(T) $, enabling inductive arguments.
• Define the family $ \mathcal{H} $ of trees (paths and star-like trees) for which $ F_t(T) = F(T) + 1 $, and prove this equality holds iff $ T \in \mathcal{H} $ via case analysis on strong support vertices.
• Construct a family of trees with arbitrarily large $ F_t(T) - F(T) $ gap by adding pendant edges to a path, showing $ F_t(T) = 2k $, $ F(T) = k $ for path of length $ k $.

Experimental results

Research questions

• RQ1 What is the maximum possible value of the total forcing number $ F_t(T) $ for a tree $ T $ of order $ n $ and maximum degree $ \Delta $, and which trees achieve this bound?
• RQ2 Which trees satisfy $ F_t(T) = n_1 $, where $ n_1 $ is the number of leaves, and how can this family be characterized?
• RQ3 Under what conditions does $ F_t(T) = F(T) + 1 $ hold for a tree $ T $, and what is the structural characterization of such extremal trees?
• RQ4 Can the difference between $ F_t(T) $ and $ F(T) $ be made arbitrarily large, and if so, how can such trees be constructed?
• RQ5 How do the invariants $ F_t(T) $ and $ F(T) $ relate in general trees, and what is the minimal possible gap between them?

Key findings

• The total forcing number of a tree $ T $ of order $ n \geq 3 $ and maximum degree $ \Delta $ satisfies $ F_t(T) \leq \frac{1}{\Delta}((\Delta-1)n + 1) $, with equality if and only if $ T $ is a specific family of trees constructed from a central vertex of degree $ \Delta $ with $ \Delta-1 $ paths of equal length attached.
• For any tree $ T $, $ F_t(T) \geq n_1 $, where $ n_1 $ is the number of leaves, and equality holds if and only if $ T $ is a path or $ \mathrm{trim}(T) \cong K_{1,n-1} $, i.e., $ T \in \mathcal{H} $.
• The total forcing number is strictly greater than the standard forcing number: $ F_t(T) \geq F(T) + 1 $ for all non-trivial trees $ T $, and equality holds if and only if $ T \in \mathcal{H} $.
• The family $ \mathcal{H} $ consists of all trees that are either paths or have a trimmed tree isomorphic to a star $ K_{1,n-1} $, and these are the only trees achieving the minimal gap between $ F_t(T) $ and $ F(T) $.
• For every integer $ k \geq 1 $, there exists a tree $ T $ such that $ F_t(T) = F(T) + k $, constructed by adding two pendant edges to each vertex of a path of length $ k $, yielding $ F_t(T) = 2k $, $ F(T) = k $.