A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques. Abu-Khzam (2017) introduced the $(a,d)$-{Cluster Editing} problem, where for fixed natural numbers $a,d$, given a graph $G$ and vertex-weights $a^*:\ V(G)\rightarrow \{0,1,\dots, a\}$ and $d^*{}:\ V(G)\rightarrow \{0,1,\dots, d\}$, we are to decide whether $G$ can be turned into a cluster graph by deleting at most $d^*(v)$ edges incident to every $v\in V(G)$ and adding at most $a^*(v)$ edges incident to every $v\in V(G)$. Results by Komusiewicz and Uhlmann (2012) and Abu-Khzam (2017) provided a dichotomy of complexity (in P or NP-complete) of $(a,d)$-{Cluster Editing} for all pairs $a,d$ apart from $a=d=1.$ Abu-Khzam (2017) conjectured that $(1,1)$-{Cluster Editing} is in P. We resolve Abu-Khzam's conjecture in affirmative by (i) providing a serious of five polynomial-time reductions to $C_3$-free and $C_4$-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving $(1,1)$-{Cluster Editing} on $C_3$-free and $C_4$-free graphs of maximum degree at most 3.

翻译：图$H$是团图当且仅当$H$是若干团的不交并。Abu-Khzam（2017）提出了$(a,d)$-簇编辑问题：对于固定的自然数$a,d$，给定图$G$以及顶点权重$a^*:\ V(G)\rightarrow \{0,1,\dots, a\}$和$d^*{}:\ V(G)\rightarrow \{0,1,\dots, d\}$，需要判断能否通过删除每个顶点$v\in V(G)$至多$d^*(v)$条关联边并添加每个顶点$v\in V(G)$至多$a^*(v)$条关联边，将$G$转化为一个簇图。Komusiewicz与Uhlmann（2012）及Abu-Khzam（2017）的研究指出，除$a=d=1$外，$(a,d)$-簇编辑问题对所有参数对$(a,d)$存在复杂性二分性（属于P类或NP完全）。Abu-Khzam（2017）猜想$(1,1)$-簇编辑问题属于P类。我们通过以下步骤肯定地解决了这一猜想：（i）通过一系列五次多项式时间归约，将问题约化为最大度至多为3的无$C_3$且无$C_4$的图；（ii）设计了一个多项式时间算法，用于求解最大度至多为3的无$C_3$且无$C_4$图上的$(1,1)$-簇编辑问题。