狄拉克定理的算法扩展
Algorithmic Extensions of Dirac's Theorem
AI总结：
本研究针对狄拉克定理相关的两个算法未解问题，证明了2-连通图中除k个顶点外其余顶点度≥δ时，判定是否存在长度至少为min{2δ+k,n}的环可在2^O(k)·n^O(1)时间内完成，推广了狄拉克定理的算法版本。
摘要：
1952年，狄拉克证明了关于具有大最小顶点度的图中长环的如下定理：每个顶点数为$n$、最小顶点度$\boldsymbol{\text{δ}\boldsymbol{\text{≥}2}$的2-连通图$G$，都包含一个长度至少为$\boldsymbol{\text{min}\boldsymbol{\textbf{2δ},n\textbf{}}$的环。特别地，若$\boldsymbol{\text{δ}\boldsymbol{\text{≥}n/2}$，则$G$是哈密顿图。狄拉克定理的证明是构造性的，可推导出一种在多项式时间内计算对应环的算法。狄拉克定理的组合界是紧的，原因如下：存在一些2-连通图，其环的长度不超过$2δ+1$；也存在一些非哈密顿图，除一个顶点外，其余所有顶点的度都至少为$n/2$。这自然引出了以下算法问题：对于$k\boldsymbol{\text{≥}1}$，（A）判定一个2-连通图是否包含长度至少为$\boldsymbol{\text{min}\boldsymbol{\textbf{2δ}+k,n\textbf{}}$的环，难度有多大？（B）当图$G$中至少有$n-k$个顶点的度至少为$n/2-k$时，判定$G$是否为哈密顿图，难度有多大？第一个问题由Fomin、Golovach、Lokshtanov、Panolan、Saurabh和Zehavi提出，第二个问题由Jansen、Kozma和Nederlof提出。即使在$k=1$的极特殊情况下，是否存在多项式时间算法来判定$G$是否包含长度至少为$\boldsymbol{\text{min}\boldsymbol{\textbf{2δ}+1,n\textbf{}}$的环，此前仍是未解之谜。我们通过证明狄拉克定理的如下算法推广解决了这两个问题：若2-连通图$G$中除$k$个顶点外，其余所有顶点的度都至少为$δ$，则判定$G$是否存在长度至少为$\boldsymbol{\text{min}\boldsymbol{\textbf{2δ}+k,n\textbf{}}$的环，可在$2^{\boldsymbol{\text{O}}(k)}\boldsymbol{\text{·}}n^{\boldsymbol{\text{O}}(1)}$的时间内完成。该狄拉克定理算法推广的证明建立在新的图论结果之上，这些结果本身也具有研究价值。
英文摘要：
In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every $n$-vertex $2$-connected graph $G$ with minimum vertex degree $δ\geq 2$ contains a cycle with at least $\min\{2δ,n\}$ vertices. In particular, if $δ\geq n/2$, then $G$ is Hamiltonian. The proof of Dirac's theorem is constructive, and it yields an algorithm computing the corresponding cycle in polynomial time. The combinatorial bound of Dirac's theorem is tight in the following sense. There are 2-connected graphs that do not contain cycles of length more than $2δ+1$. Also, there are non-Hamiltonian graphs with all vertices but one of degree at least $n/2$. This prompts naturally to the following algorithmic questions. For $k\geq 1$, (A) How difficult is to decide whether a 2-connected graph contains a cycle of length at least $\min\{2δ+k,n\}$? (B) How difficult is to decide whether a graph $G$ is Hamiltonian, when at least $n - k$ vertices of $G$ are of degrees at least $n/2-k$? The first question was asked by Fomin, Golovach, Lokshtanov, Panolan, Saurabh, and Zehavi. The second question is due to Jansen, Kozma, and Nederlof. Even for a very special case of $k=1$, the existence of a polynomial-time algorithm deciding whether $G$ contains a cycle of length at least $\min\{2δ+1,n\}$ was open. We resolve both questions by proving the following algorithmic generalization of Dirac's theorem: If all but $k$ vertices of a $2$-connected graph $G$ are of degree at least $δ$, then deciding whether $G$ has a cycle of length at least $\min\{2δ+k, n\}$ can be done in time $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$. The proof of the algorithmic generalization of Dirac's theorem builds on new graph-theoretical results that are interesting on their own.
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