For $t,g>0$, a vertex-weighted graph of total weight $W$ is $(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is trimmable if for each constant $t>0$, there is a constant $g=g(t)$ such that every vertex-weighted graph in the family is $(t,g)$-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. Based on this result, we derive a polynomial-time approximation scheme for the problem of labeling weighted points with nonoverlapping sliding labels of unit height and given lengths so as to maximize the total weight of the labeled points. This settles one of the last major open questions in the theory of map labeling.
Path covering number and L(2,1)-labeling number of graphs
图$G$的{ \it路覆盖}是$G$的顶点不相交路的集合，包含$G$的所有顶点。由$P(G)$表示的$G$的{ \it路径覆盖数}是$G$的路径覆盖中的路径的最小数目。图$G$的{ \sl $k$ -L（2，1）-标号}是从$V(G)$到集合${0,1,...,k}$的映射$f$，使得如果$d_G(u,v)=1$则为$|f(u)-f(v)|\ge 2$，如果$d_G(u,v)=2$则为$|f(u)-f(v)|\ge 1$. $G$的{ \sl L（2，1）-标记数$λ(G)$ }是使得$G$具有$k$-L（2，1）-标记的最小数$k$.研究图的路覆盖数和L（2，1）-标号数。我们的主要工作扩展了[On island sequences of labelings with a condition at distance two，Discrete Applied Maths 158（2010），1-7]中的大部分结果，并且可以回答[On the structure of graphs with non-surjective L（2，1）-labelings，SIAM J. Discrete Math.19（2005），208-223]中的一个公开问题。
