Signed double Roman domination on cubic graphs

TL;DR

This work first considers the problem on general cubic graphs of order n, and presents a sharp $n/2+\Theta(1)$ lower bound for the SDRDN by means of the discharging method, and derives a new best upper bound.

Abstract

The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from $\{\pm{}1,2,3\}$ to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility is given whenever (i) vertices labeled $\pm{}1$ have at least one neighbor with label in $\{2,3\}$; (ii) each vertex labeled $-1$ has one $3$-labeled neighbor or at least two $2$-labeled neighbors; and (iii) the sum of labels over the closed neighborhood of any vertex is positive. The cumulative weight of an optimal labeling is called signed double Roman domination number (SDRDN). In this work, we first consider the problem on general cubic graphs of order $n$ for which we present a sharp $n/2+Θ(1)$ lower bound for the SDRDN by means of the discharging method. Moreover, we derive a new best upper bound. Observing that we are often able to minimize the SDRDN over the class of cubic graphs of a fixed order, we then study in this context generalized Petersen graphs for independent interest, for which we propose a constraint programming guided proof. We then use these insights to determine the SDRDNs of subcubic $2\times m$ grid graphs, among other results.

Figures (8)

• Figure 1: Exemplary graphs for the special graph classes considered in this work.
• Figure 2: Optimal SDRDFs for $P_{m,3}$ when $m=4\ell+1$ respectively $m=4\ell+3$. In both cases, a label pattern of width $4$ is periodically repeated $\ell-1$ respectively $\ell-2$ times to finally be flanked by a termination pattern of width $5$ respectively $11$. The labeling is exemplarily illustrated for $m=13$ respectively $m=19$.
• Figure 3: The graph $G'$ in \ref{['fig:two-by-eight-grid-graph-after-cutoff']} is the result of deleting four of the vertical edges from $G$ in \ref{['fig:two-by-twelve-grid-graph']} and successively performing eight edge contractions.
• Figure 4: Schemes for optimal labelings given in Theorem \ref{['thm:generalized-petersen-graph-case-k-equal-one']} for the graph $P_{m,1}$.
• Figure 5: Extending $G_{2,m}$ to a cubic graph via different constructions.

Theorems & Definitions (26)

• Theorem 1: amjadi2018signed
• Theorem 2: henning2012alpha
• Proposition 1
• proof
• proof
• Theorem 3
• proof
• Remark 1
• Theorem 4
• proof