远离简并
Going Far From Degeneracy
摘要：
一个无向图G是d-退化的，如果G的每个子图都有一个度至多为d的顶点.根据1959年Erdés和Gallai的经典定理，每个退化度d>1的图都包含一个长度至少为d+1的圈。Erdés和Gallai的证明是有建设性的，可以转化为一个多项式时间算法，构造一个长度至少为d+1的循环。但是我们能在多项式时间内判定一个图是否包含一个长度至少为d+2的圈吗？从哈密尔顿圈的一个简单的约化给出了这个问题的否定答案：判定一个图是否有长度至少为d+2的圈是NP-完全的。令人惊讶的是，当输入图是2-连通的时，问题的复杂性急剧变化。在这种情况下，我们证明了判定G是否包含长度至少为d+k的圈可以在时间2^{O（k）}内完成。|V（G）|时间复杂度为O（1）换句话说，判定一个2-连通n-顶点G是否包含长度至少为d+log n的圈可以在多项式时间内完成。类似的算法结果适用于图中的长路径。我们观察到，决定一个图是否有一条长度至少为d+1的路是NP-完全的。然而，我们证明了，如果图G是连通的，那么判定G是否包含长度至少为d+k的路可以在时间2^{O（k）}n^{O（1）}内完成。我们通过证明选择退化度作为“上述保证参数化”在以下意义上是最优的来补充这些结果：对于任何ε>0，判定退化度为d的连通（2-连通）图是否有长度至少为（1+ε）d的路（圈）是NP-完全的。
英文摘要：
An undirected graph G is d-degenerate if every subgraph of G has a vertex of degree at most d. By the classical theorem of Erdős and Gallai from 1959, every graph of degeneracy d>1 contains a cycle of length at least d+1. The proof of Erdős and Gallai is constructive and can be turned into a polynomial time algorithm constructing a cycle of length at least d+1. But can we decide in polynomial time whether a graph contains a cycle of length at least d+2? An easy reduction from Hamiltonian Cycle provides a negative answer to this question: deciding whether a graph has a cycle of length at least d+2 is NP-complete. Surprisingly, the complexity of the problem changes drastically when the input graph is 2-connected. In this case we prove that deciding whether G contains a cycle of length at least d+k can be done in time 2^{O(k)}|V(G)|^{O(1)}. In other words, deciding whether a 2-connected n-vertex G contains a cycle of length at least d+log n can be done in polynomial time. Similar algorithmic results hold for long paths in graphs. We observe that deciding whether a graph has a path of length at least d+1 is NP-complete. However, we prove that if graph G is connected, then deciding whether G contains a path of length at least d+k can be done in time 2^{O(k)}n^{O(1)}. We complement these results by showing that the choice of degeneracy as the `above guarantee parameterization' is optimal in the following sense: For any ε>0 it is NP-complete to decide whether a connected (2-connected) graph of degeneracy d has a path (cycle) of length at least (1+ε)d.
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