SIAM Journal on Discrete Mathematics, Volume 34, Issue 2, Page 1302-1325, January 2020.
A graph $G$ is contractible to a graph $H$ if there is a set $X \subseteq E(G)$, such that $G/X$ is isomorphic to $H$. Here, $G/X$ is the graph obtained from $G$ by contracting all the edges in $X$. For a family of graphs $\cal F$, the $\mathcal{F}$-Contraction problem takes as input a graph $G$ on $n$ vertices, and the objective is to output the largest integer $t$, such that $G$ is contractible to a graph $H \in {\cal F}$, where $|V(H)|=t$. When $\cal F$ is the family of paths, then the corresponding $\mathcal{F}$-Contraction problem is called Path Contraction. The problem Path Contraction admits a simple algorithm running in time $2^{n}\cdot n^{{\mathcal{O}}(1)}$. In spite of the deceptive simplicity of the problem, beating the $2^{n}\cdot n^{{\mathcal{O}}(1)}$ bound for Path Contraction seems quite challenging. In this paper, we design an exact exponential time algorithm for Path Contraction that runs in time ${{1.99987}^n}\cdot n^{{\mathcal O}(1)}$. We also define a problem called 3-Disjoint Connected Subgraphs and design an algorithm for it that runs in time $1.88^n\cdot n^{{\mathcal O}(1)}$. The above algorithm is used as a subroutine in our algorithm for Path Contraction.

中文翻译：

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路径收缩快于$ 2 ^ n $

SIAM离散数学杂志，第34卷，第2期，第1302-1325页，2020年1月。
如果存在一组$ X \ subseteq E（G）$，则图$ G $会收缩为图$ H $，因此$ G / X $与$ H $同构。这里，$ G / X $是通过收缩$ X $中的所有边而从$ G $获得的图形。对于一族图形$ \ cal F $，$ \ mathcal {F} $-Contract问题将$ n $顶点上的图形$ G $作为输入，目标是输出最大的整数$ t $，例如$ G $可收缩为{\ cal F} $中的图$ H \，其中$ | V（H）| = t $。当$ \ cal F $是路径族时，相应的$ \ mathcal {F} $-收缩问题称为路径收缩。路径收缩问题允许在时间$ 2 ^ {n} \ cdot n ^ {{\ mathcal {O}}（1）} $中运行一个简单算法。尽管问题的方法很简单，但击败路径收缩约束$ 2 ^ {n} \ cdot n ^ {{\ mathcal {O}}（1）} $似乎很有挑战性。在本文中，我们设计了路径收缩的精确指数时间算法，该算法在时间$ {{{1.99987} ^ n} \ cdot n ^ {{\\ mathcal O}（1）} $中运行。我们还定义了一个称为3-Disjoint Connected Subgraphs的问题，并为其设计了一种算法，该算法可在$ 1.88 ^ n \ cdot n ^ {{\\ mathcal O}（1）} $的时间运行。上述算法在我们的路径收缩算法中用作子例程。