Open Packing in Graphs: Bounds and Complexity

TL;DR

The paper conducts a comprehensive complexity analysis of OPEN PACKING and its dual ρ^o(G) across H-free graphs and split-graph subclasses. It delivers a sharp H-free dichotomy: Open Packing is polynomial-time solvable on H-free graphs if H ∈ {pK_1,(K_2∪(p-2)K_1),(P_3∪(p-3)K_1),(P_4∪(p-4)K_1)}; otherwise NPC, with a separate NPC result for K_{1,3}-free graphs and a polynomial tractability for (P_4∪rK_1)-free graphs for all r≥1. The work also establishes tight structural bounds such as ρ^o(G) ≤ 2r+1 and γ_t(G) ≤ 2r+2 for connected (P_4∪rK_1)-free graphs, and presents similar dichotomies in split graphs and I_r-split graphs, including NP-completeness results for certain r and polynomial-time solvability for small r. Collectively, the results illuminate the relationship between Open Packing and Total Domination (γ_t) under restricted graph classes, reveal parameterized hardness and approximation limits (e.g., W[1]-hardness on K_{1,3}-free graphs), and provide polynomial-time algorithms in key subclasses. These findings contribute precise boundaries for when efficient computation is possible and when intractability persists in graph packing/dominating problems.

Abstract

Given a graph $G(V,E)$, a vertex subset $S$ of $G$ is called an open packing in $G$ if no pair of distinct vertices in $S$ have a common neighbour in $G$. The size of a largest open packing in $G$ is called the open packing number, $ρ^o(G)$, of $G$. It would be interesting to note that the open packing number is a lower bound for the total domination number in graphs with no isolated vertices [Henning and Slater, 1999]. Given a graph $G$ and a positive integer $k$, the decision problem OPEN PACKING tests whether $G$ has an open packing of size at least $k$. The optimization problem MAX-OPEN PACKING takes a graph $G$ as input and finds the open packing number of $G$. It is known that OPEN PACKING is NP-complete on split graphs (i.e., $\{2K_2,C_4,C_5\}$-free graphs) [Ramos et al., 2014]. In this work, we complete the study on the complexity (P vs NPC) of OPEN PACKING on $H$-free graphs for every graph $H$ with at least three vertices by proving that OPEN PACKING is (i) NP-complete on $K_{1,3}$-free graphs and (ii) polynomial time solvable on $(P_4\cup rK_1)$-free graphs for every $r\geq 1$. In the course of proving (ii), we show that for every $t\in {2,3,4}$ and $r\geq 1$, if G is a $(P_t\cup rK_1)$-free graph, then $ρ^o(G)$ is bounded above by a linear function of $r$. Moreover, we show that OPEN PACKING parameterized by solution size is W[1]-complete on $K_{1,3}$-free graphs and MAX-OPEN PACKING is hard to approximate within a factor of $n^{(\frac{1}{2}-δ)}$ for any $δ>0$ on $K_{1,3}$-free graphs unless P=NP. Further, we prove that OPEN PACKING is (a) NP-complete on $K_{1,4}$-free split graphs and (b) polynomial time solvable on $K_{1,3}$-free split graphs. We prove a similar dichotomy result on split graphs with degree restrictions on the vertices in the independent set of the clique-independent set partition of the split graphs.

Figures (10)

• Figure 1: (a) a simple graph $G$, (b) $K_{1,3}$-free graph $G'$ obtained from $G$ using Construction \ref{['construct-op-id-k_1,3']}.
• Figure 2: (a) an edge $e=u_\ell u_j$ of the input graph $G$ of Construction \ref{['construct-op-id-k_1,3']} and (b) the three vertex path $u_\ell eu_j$ produced by Construction \ref{['construct-op-id-k_1,3']} along with two vertices $y,z$ of $G'$ such that $y\in N_{G'}(z)\cap N_{G'}(u_\ell)$ as in the contrary assumption of Claim \ref{['claim-eu']}. In figure, dashed lines represent the edges not in $G'$ as observed in the proof of Claim \ref{['claim-eu']}.
• Figure 3: A connected ($P_4\cup rK_1$)-free graph $G$ with $D=\{x_1,x_2,x_3,x_4,y_1,$$y_2,\ldots,y_{r-1}\}$$\subseteq V(G)$ such that $G[D]\cong (P_4\cup (r-1)K_1)$. It is possible that $w_i=w_j$ for $i\neq j$.
• Figure 4: A ($P_4\cup 3K_1$)-free graph $G_3$ defined in Remark \ref{['rem-tight-p4-rk1']} with an open packing $S_3=\{x_1,x_2,x_3,y_1,y_2,y_3,v\}$ of size $7=(2(3)+1)$. Also, $\gamma_t(G_3)=7$ with $D=\{u,z_1,y_1,z_2,y_2,z_3,y_3\}$ as a total dominating set in $G_3$.
• Figure 5: A connected $7K_1$-free graph $H_7$ with an open packing (as well as a total dominating set) $S_7=\{x_1,x_2,x_3,$$x_4,x_5,y_1,y_2,y_3,y_4,y_5\}$ of size $2(7-2)=10$. Note that $H_7$ is also a ($K_2\cup 6K_1$)-free graph.

Theorems & Definitions (75)

• Theorem 1
• Theorem 2
• proof
• Remark 1
• Claim 1
• Claim 2
• Claim 3
• Claim 4
• Claim 5
• Theorem 3