arXiv subjects

Changhong Lu

Publications and source records attributed to Changhong Lu.

25 records · Page 2Linked to original sources

arXiv2019

A Note on Roman \{2\}-domination problem in graphs

For a graph $G=(V,E)$, a Roman $\{2\}$-dominating function (R2DF)$f:V\rightarrow \{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either there exists a neighbor $u\in N(v)$, with $f(u)=2$, or at least two neighbors $x,y\in N(v)$ having $f(x)=f(y)=1$. The weight of a R2DF is the sum $f(V)=\sum_{v\in V}{f(v)}$, and the minimum weight of a R2DF is the Roman $\{2\}$-domination number $γ_{\{R2\}}(G)$. A R2DF is independent if the set of vertices having positive function values is an independent set. The independent Roman $\{2\}$-domination number $i_{\{R2\}}(G)$ is the minimum weight of an independent Roman $\{2\}$-dominating function on $G$. In this paper, we show that the decision problem associated with $γ_{\{R2\}}(G)$ is NP-complete even when restricted to split graphs. We design a linear time algorithm for computing the value of $i_{\{R2\}}(T)$ for any tree $T$. This answers an open problem raised by Rahmouni and Chellali [Independent Roman $\{2\}$-domination in graphs, Discrete Applied Mathematics 236 (2018), 408-414]. Chellali, Haynes, Hedetniemi and McRae \cite{chellali2016roman} have showed that Roman $\{2\}$-domination number can be computed for the class of trees in linear time. As a generalization, we present a linear time algorithm for solving the Roman $\{2\}$-domination problem in block graphs.

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arXiv2018

Power domination in regular claw-free graphs

In this paper, we first show that the power domination number of a connected $4$-regular claw-free graph on $n$ vertices is at most $\frac{n+1}{5}$, and the bound is sharp. The statement partly disprove the conjecture presented by Dorbec et al. in SIAM J. Discrete Math., 27:1559-1574, 2013. Then we present a dynamic programming style linear-time algorithm for weighted power domination problem in trees.

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arXiv2016

Monochromatic loose path partitions in k-uniform hypergraphs

A conjecture of Gyárfás and Sárközy says that in every $2$-coloring of the edges of the complete $k$-uniform hypergraph $K_n^k$, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most $k-2$ vertices. A weaker form of this conjecture with $2k-5$ uncovered vertices instead of $k-2$ is proved, thus the conjecture holds for $k=3$. The main result of this paper states that the conjecture is true for all $k\ge 3$.

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arXiv2012

Path covering number and L(2,1)-labeling number of graphs

A {\it path covering} of a graph $G$ is a set of vertex disjoint paths of $G$ containing all the vertices of $G$. The {\it path covering number} of $G$, denoted by $P(G)$, is the minimum number of paths in a path covering of $G$. An {\sl $k$-L(2,1)-labeling} of a graph $G$ is a mapping $f$ from $V(G)$ to the set ${0,1,...,k}$ such that $|f(u)-f(v)|\ge 2$ if $d_G(u,v)=1$ and $|f(u)-f(v)|\ge 1$ if $d_G(u,v)=2$. The {\sl L(2,1)-labeling number $λ(G)$} of $G$ is the smallest number $k$ such that $G$ has a $k$-L(2,1)-labeling. The purpose of this paper is to study path covering number and L(2,1)-labeling number of graphs. Our main work extends most of results in [On island sequences of labelings with a condition at distance two, Discrete Applied Maths 158 (2010), 1-7] and can answer an open problem in [On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005), 208-223].

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arXiv2010

On the number of a SDRs of a valued (t,n)-family

A system of distinct representatives (SDR) of a family $F = (A_1, \cdots, A_n)$ is a sequence $(x_1, \cdots, x_n)$ of $n$ distinct elements with $x_i \in A_i$ for $1 \le i \le n$. Let $N(F)$ denote the number of SDRs of a family $F$; two SDRs are considered distinct if they are different in at least one component. For a nonnegative integer $t$, a family $F=(A_1,\cdots,A_n)$ is called a $(t,n)$-family if the union of any $k\ge 1$ sets in the family contains at least $k+t$ elements. The famous Hall's Theorem says that $N(F)\ge 1$ if and only if $F$ is a $(0,n)$-family. Denote by $M(t,n)$ the minimum number of SDRs in a $(t,n)$-family. The problem of determining $M(t,n)$ and those families containing exactly $M(t,n)$ SDRs was first raised by Chang [European J. Combin.{\bf 10}(1989), 231-234]. He solved the cases when $0\le t\le 2$ and gave a conjecture for $t\ge 3$. In this paper, we solve the conjecture. In fact, we get a more general result for so-called valued $(t,n)$-family.

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arXiv2009

Vertices in all minimum paired-dominating sets of block graphs

Let $G=(V,E)$ be a simple graph without isolated vertices. A set $S\subseteq V$ is a paired-dominating set if every vertex in $V-S$ has at least one neighbor in $S$ and the subgraph induced by $S$ contains a perfect matching. In this paper, we present a linear-time algorithm to determine whether a given vertex in a block graph is contained in all its minimum paired-dominating sets.

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arXiv2009

Identifying codes and locating-dominating sets on paths and cycles

Let $G=(V,E)$ be a graph and let $r\ge 1$ be an integer. For a set $D \subseteq V$, define $N_r[x] = \{y \in V: d(x, y) \leq r\}$ and $D_r(x) = N_r[x] \cap D$, where $d(x,y)$ denotes the number of edges in any shortest path between $x$ and $y$. $D$ is known as an $r$-identifying code ($r$-locating-dominating set, respectively), if for all vertices $x\in V$ ($x \in V\backslash D$, respectively), $D_r(x)$ are all nonempty and different. In this paper, we provide complete results for $r$-identifying codes in paths and odd cycles; we also give complete results for 2-locating-dominating sets in cycles.

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