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On The Secure Domination of Mycielskian Graphs

Kiran R. Bhutani, Anthony Christiana, Peter Ulrickson

Abstract

We study the secure domination number of the Mycielskian graph of a simple, connected, undirected graph. We give generally applicable bounds, compute secure domination numbers for Mycielskians of important families of graphs, and construct families of graphs realizing particular values of (secure) domination parameters.

On The Secure Domination of Mycielskian Graphs

Abstract

We study the secure domination number of the Mycielskian graph of a simple, connected, undirected graph. We give generally applicable bounds, compute secure domination numbers for Mycielskians of important families of graphs, and construct families of graphs realizing particular values of (secure) domination parameters.
Paper Structure (14 sections, 19 theorems, 12 equations, 4 figures)

This paper contains 14 sections, 19 theorems, 12 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Secure Domination
  3. The Mycielskian of a Graph
  4. General Bounds
  5. Dominating Vertices
  6. $S$-isolates
  7. Ordinary Domination Bound
  8. Special Families
  9. Paths and Cycles
  10. Complete Bipartite Graphs
  11. Constructions for given parameters
  12. Prescribed Gaps
  13. Prescribed Values
  14. Future Work

Key Result

Theorem 1.1

ck$\gamma_s(G) =1$ if and only if $G$ is a complete graph.

Figures (4)

  • Figure 1: The Mycieslkian of $P_4$
  • Figure 2: A dominating set in $P_5$ and private neighbors of $w$
  • Figure 3: A good dominating set for $P_7$ that is inefficient for $\mu(P_7)$.
  • Figure 4: Pairing guards in $P_7$ yields an efficient secure dominating set for $\mu(P_7)$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Definition 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 1.1
  • proof
  • Corollary 1.1
  • proof
  • ...and 30 more