A vertex v is a Grundy vertex with respect to a proper k-coloring c of a graph $G=(V,E)$ if v has a neighbor of color j for every j $(1\le j<i\le k)$, where $i=c(v)$. A proper k-coloring c of G is called a Grundy k-coloring of G if every vertex is a Grundy vertex with respect to c and the largest integer k such that G admits a Grundy k-coloring is called the Grundy number of G which is denoted as $\mathrm{\Gamma }(G)$. Given a graph G and an integer k, the Grundy number decision problem is to decide whether $\mathrm{\Gamma }(G)\ge k$. The Grundy number decision problem is known to be NP-complete for bipartite graphs and complement of bipartite graphs. In this paper, we strengthen this result by showing that this problem remains NP-complete for perfect elimination bipartite graphs as well as for complement of perfect elimination bipartite graphs. Further, we give a linear-time algorithm to find the Grundy number of chain graphs, which is a proper subclass of the class of perfect elimination bipartite graphs. We also give a linear-time algorithm to find the Grundy number in complements of chain graphs. A partial Grundy coloring of a graph G is a proper k-coloring of G such that there is at least one Grundy vertex with each color i, $1\le i\le k$ and the partial Grundy number of G, $\partial \mathrm{\Gamma }(G)$, is the largest integer k such that G admits a partial Grundy k-coloring. Given a graph G and an integer k, the partial Grundy number decision problem is to decide whether $\partial \mathrm{\Gamma }(G)\ge k$. It is known that the partial Grundy number decision problem is NP-complete for bipartite graphs. In this paper, we prove that this problem is NP-complete in the complements of bipartite graphs by showing that the Grundy number and partial Grundy number are equal in complements of bipartite graphs.

顶点v是关于图的适当k着色c的Grundy顶点$G=（V，Ë）$如果v每j有一个颜色j的邻居 $（\mathrm{1个}\le Ĵ<\mathrm{一世}\le ķ）$，在哪里 $\mathrm{一世}=C（v）$。一个适当的ķ -coloring Ç的ģ称为格伦迪K-着色的ģ如果每个顶点是相对于一个顶点格伦迪ç和最大整数ķ使得ģ承认一个格伦迪ķ -coloring称为格伦迪数目的ģ其中表示为$\mathrm{\Gamma }（G）$。给定一个图G和一个整数k，格兰迪数决定问题是决定是否$\mathrm{\Gamma }（G）\ge ķ$。对于二部图和二部图的补图，已知格伦迪数决策问题是NP完全的。在本文中，我们通过证明对于完全消除二分图以及对于完美消除二分图，该问题仍然是NP完全的，从而加强了这一结果。此外，我们给出了一个线性时间算法来找到链图的Grundy数，它是理想消除二分图类的适当子类。我们还给出了线性时间算法，以在链图的补码中找到格伦迪数。的曲线图的部分格伦迪着色ģ是一个适当的ķ -coloring的ģ使得存在至少一个顶点格伦迪与每个颜色我，$\mathrm{1个}\le \mathrm{一世}\le ķ$和局部格伦迪数目的ģ，$\partial \mathrm{\Gamma }（G）$是最大的整数k，以使G允许部分Grundy k着色。给定一个图G和一个整数k，偏葛兰迪数决定问题是决定是否$\partial \mathrm{\Gamma }（G）\ge ķ$。众所周知，对于二部图，部分Grundy数决策问题是NP完全的。在本文中，通过证明二部图的补中的格朗迪数和部分格伦迪数相等，证明了该问题在二部图的补图中是NP完全的。