区间图上的1-固定端点路径覆盖问题是多项式问题
The 1-fixed-endpoint Path Cover Problem is Polynomial on Interval Graph
摘要：
我们考虑一个变种的路径覆盖问题，即$k$固定端点路径覆盖问题，或简称kPC，区间图。给定图$G$和$V(G)$的$k$顶点的子集$\mathcal{T}$，$G$关于$\mathcal{T}$的$k$ -固定端点路径覆盖是覆盖$G$的顶点的顶点不相交路径$\mathcal{P}$的集合，使得$\mathcal{T}$的$k$顶点是$\mathcal{P}$中的路径的所有端点。kPC问题是找到$k$-最小基数的$G$的固定端点路径覆盖;注意，如果$\mathcal{T}$为空，则所述问题与经典路径覆盖问题一致。本文研究了区间图上的1-固定端点路径覆盖问题，简称1 PC，推广了1HP问题，该问题已被证明即使在小类图上也是NP-完全的。出于工作的Damaschke，在那里他离开了1HP和2 HP问题的类的区间图，我们表明，1 PC问题可以解决在多项式时间的类的区间图。该算法简单，在$O(n^2)$时间内运行，所需空间为线性空间，并且能够在相同的时间和空间复杂度内求解区间图上的1HP问题.
英文摘要：
We consider a variant of the path cover problem, namely, the $k$-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph $G$ and a subset $\mathcal{T}$ of $k$ vertices of $V(G)$, a $k$-fixed-endpoint path cover of $G$ with respect to $\mathcal{T}$ is a set of vertex-disjoint paths $\mathcal{P}$ that covers the vertices of $G$ such that the $k$ vertices of $\mathcal{T}$ are all endpoints of the paths in $\mathcal{P}$. The kPC problem is to find a $k$-fixed-endpoint path cover of $G$ of minimum cardinality; note that, if $\mathcal{T}$ is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke, where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. The proposed algorithm is simple, runs in $O(n^2)$ time, requires linear space, and also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
