Since Jacobson [FOCS89] initiated the investigation of succinct graph encodings 35 years ago, there has been a long list of results on balancing the generality of the class, the speed, the succinctness of the encoding, and the query support. Let Cn denote the set consisting of the graphs in a class C that with at most n vertices. A class C is nontrivial if the information-theoretically min number log |Cn| of bits to distinguish the members of Cn is Omega(n). An encoding scheme based upon a single class C is C-opt if it takes a graph G of Cn and produces in deterministic O(n) time an encoded string of at most log |Cn| + o(log |Cn|) bits from which G can be recovered in O(n) time. Despite the extensive efforts in the literature, trees and general graphs were the only nontrivial classes C admitting C-opt encoding schemes that support the degree query in O(1) time. Basing an encoding scheme upon a single class ignores the possibility of a shorter encoded string using additional properties of the graph input. To leverage the inherent structures of individual graphs, we propose to base an encoding scheme upon of multiple classes: An encoding scheme based upon a family F of classes, accepting all graphs in UF, is F-opt if it is C-opt for each C in F. Having a C-opt encoding scheme for each C in F does not guarantee an F-opt encoding scheme. Under this more stringent criterion, we present an F-opt encoding scheme for a family F of an infinite number of classes such that UF comprises all graphs of bounded Hadwiger numbers. F consists of the nontrivial quasi-monotone classes of k-clique-minor-free graphs for each positive integer k. Our F-opt scheme supports queries of degree, adjacency, neighbor-listing, and bounded-distance shortest path in O(1) time per output. We broaden the graph classes admitting opt encoding schemes that also efficiently support fundamental queries.

翻译：自Jacobson[FOCS89]于35年前开创简洁图编码的研究以来，关于平衡类的通用性、速度、编码简洁性及查询支持的结果层出不穷。记C_n为类C中顶点数不超过n的图构成的集合。若信息论意义上区分C_n中成员所需的最小比特数log|C_n|为Omega(n)，则称类C为非平凡类。基于单一类C的编码方案称为C-最优的，若它能在确定性O(n)时间内为C_n中的图G生成至多log|C_n|+o(log|C_n|)比特的编码字符串，且G可在O(n)时间内从该字符串恢复。尽管文献中已有大量努力，但树和一般图是仅有的两类支持O(1)时间度查询的C-最优编码方案的非平凡类C。基于单一类的编码方案忽略了利用图输入的额外属性获得更短编码字符串的可能性。为利用单个图的内在结构，我们提出基于多类的编码方案：基于类族F的编码方案接受UF中所有图，若它对F中每个C都是C-最优的，则称其为F-最优的。对F中每个C存在C-最优编码方案并不能保证F-最优编码方案的存在。在此更严格的标准下，我们为包含无限多个类的类族F提出了一个F-最优编码方案，使得UF包含所有有界哈德维格数的图。F由每个正整数k对应的k-团-无 minors 图的非平凡拟单调类组成。我们的F-最优方案支持每个输出O(1)时间的度查询、邻接查询、邻居列表查询及有界距离最短路径查询。我们拓宽了支持最优编码方案且同时高效支持基本查询的图类范围。