arXiv subjects

Weiping Shang

Publications and source records attributed to Weiping Shang.

3 recordsLinked to original sources

arXiv2026

Semitotal domination in unit disk graphs

A set $S \subseteq V$ is called a {\em semitotal dominating set} of $G=(V,E)$ if every vertex in $V \setminus S$ is adjacent to at least one vertex in $S$, and every vertex in $S$ is within distance 2 of another vertex in $S$. The corresponding decision problem is NP-complete even for unit disk graphs. In this paper, we present a 5-factor approximation algorithm for the Minimum Semitotal Domination problem on unit disk graphs in the graph-based input model. The algorithm processes the layers of a Breadth-First-Search tree and constructs a maximal independent set whose vertices satisfy the semitotal condition. For a graph with $n$ vertices and $m$ edges, the algorithm runs in $O(n + m)$ time, and hence in $O(n^2)$ time in the worst case. This improves the previously known 5.75-approximation algorithm with $O(n^3)$ running time.

cs.DS↗

arXiv2026

Double Italian domination in trees

Let $G$ be a graph with vertex set $V=V(G)$. A double Roman dominating function on a graph $G$ is a function $f : V \to \{0,1,2,3\}$ satisfying the conditions that if $f(v) = 0$, then vertex $v$ must have at least two neighbors in $V_2$ or one neighbor in $V_3$, if $f(v) = 1$, then vertex $v$ must have at least one neighbor in $V_2 \cup V_3$. The weight of a double Roman dominating function $f$ is the sum $f(V) = \sum_{v \in V} f(v)$, and the double Roman domination number $\gamma_{dR}(G)$ is the minimum weight of a double Roman dominating function on $G$. A double Italian dominating function on a graph $G$ is a function $f : V \to \{0,1,2,3\}$ satisfying the condition that for every vertex $u \in V$, if $f(u) \in \{0,1\}$, then $\sum_{v \in N[u]} f(v) \ge 3$. The double Roman domination number $\gamma_{dI}(G)$ is the minimum weight of a double Italian dominating function on $G$. Mojdeh and Volkmann [D.A. Mojdeh and L. Volkmann, Roman {3}-domination (double Italian domination), Discrete Appl. Math. 283 (2020), 555--564] proved that $\gamma_{dI}(T) = \gamma_{dR}(T)$ for any tree $T$. However, we find that there is a minor issue in the proof. In this paper, we first prove that $\gamma_{dI}(T) \neq \gamma_{dR}(T)$. Subsequently, we present a sharp bound on the double Italian domination number of any non-trivial tree $T$, and characterize the trees attaining this bound.

math.CO↗

arXiv2013

A Graph Theoretical Approach to Network Encoding Complexity

Consider an acyclic directed network $G$ with sources $S_1, S_2,..., S_l$ and distinct sinks $R_1, R_2,..., R_l$. For $i=1, 2,..., l$, let $c_i$ denote the min-cut between $S_i$ and $R_i$. Then, by Menger's theorem, there exists a group of $c_i$ edge-disjoint paths from $S_i$ to $R_i$, which will be referred to as a group of Menger's paths from $S_i$ to $R_i$ in this paper. Although within the same group they are edge-disjoint, the Menger's paths from different groups may have to merge with each other. It is known that by choosing Menger's paths appropriately, the number of mergings among different groups of Menger's paths is always bounded by a constant, which is independent of the size and the topology of $G$. The tightest such constant for the all the above-mentioned networks is denoted by $\mathcal{M}(c_1, c_2,..., c_2)$ when all $S_i$'s are distinct, and by $\mathcal{M}^*(c_1, c_2,..., c_2)$ when all $S_i$'s are in fact identical. It turns out that $\mathcal{M}$ and $\mathcal{M}^*$ are closely related to the network encoding complexity for a variety of networks, such as multicast networks, two-way networks and networks with multiple sessions of unicast. Using this connection, we compute in this paper some exact values and bounds in network encoding complexity using a graph theoretical approach.

cs.IT↗