The Graph Parameter Hierarchy February 15, 2019 1 About The purpose of this document is to gather graph parameters (also called graph invariants) and their relationships in a central place. It was edited by Manuel Sorge1 and Mathias Weller2 with contributions by Ren´evan Bevern, Florent Foucaud, OndˇrejSuch´y,Pascal Ochem, Martin Vatshelle, and Gerhard J. Woeg- inger. This project and the diagram shown in Figure 1 is inspired by the work of Bart M. P. Jansen [20]. Some further related work (in no particular order) is the Information System on Graph Classes and their Inclusions (ISGCI)3, the Graph Parameter Project4, INGRID: a graph invariant manipulator [12] and the work by Martin Vatshelle [26]. 2 Terminology A graph parameter is a function φ: G ! R where G is the set of finite graphs and R is the set of real numbers. Let φ, graph parameters. The parameter φ upper bounds another parameter , if there is some function f such that for every graph G in G it holds that (G) ≤ f(φ(G)); we write φ . Parameter φ is unbounded in if :( φ). Parameter φ strictly upper bounds if φ ^ :( φ). If (:(φ )) ^ :( φ) then φ and are incomparable. If φ ^ φ then φ and are equal. [email protected] [email protected] 3http://www.graphclasses.org 4https://robert.sasak.sk/gpp/index.php 1 Distance to Clique Vertex Cover Max Leaf # 4.28 4.10 4.25 Minimum Distance to Distance to Distance to Feedback Treedepth Clique Cover Co-Cluster Cluster Disjoint Paths Edge Set Bandwidth 4.26 4.22 4.27 Maximum Distance to Distance to Feedback Maximum Bisection Independent Set Cograph Interval Vertex Set Pathwidth 4.19 Degree Width 4.20 Minimum Distance to Distance to Dominating Set Chordal Outerplanar Genus h-index 4.17 4.16 Max Diameter 4.8 4.7 2 of Components Treewidth 4.21 as 4.16 4.12 Average Acyclic Distance Chromatic # 4.23 4.9 Boxicity 4.13 4.18 Girth 4.15 Degeneracy Distance to Cliquewidth Chordality Bipartite Average Degree Distance to Perfect Minimum Chromatic # Degree Maximum Distance to Domatic # Clique Disconnected Figure 1: Hasse diagram of the known part of the boundedness relation between graph parameters. 3 Parameter Definitions 3.1 Acyclic Chromatic Number The acyclic chromatic number of a graph G = (V; E) is the smallest size of a vertex partition P = fV1;:::;V`g such that each Vi is an independent set and for all Vi;Vj the graph G[Vi [ Vj] does not contain a cycle. In other words, the acyclic chromatic number is the smallest number of colors needed for a proper vertex coloring such that every two-chromatic subgraph is acyclic. Introduced by Gr¨unbaum [18]. 3.2 Covering Parameters 3.2.1 Path Number The path number of a graph G is the minimum number of paths the edges of G can be partitioned into [2]. Exists in \disjoint" and \overlapping" versions where the paths have to be disjoint or not, respectively. 3.2.2 Arboricity The arboricity of a graph G is the minimum number of forests the edges of G can be partitioned into. It is called linear arboricity if the forests are linear (collection of paths). 3.2.3 Vertex Arboricity The vertex arboricity (or \point arboricity") of a graph G is the minimum number of vertex subsets Vi of G such that G[Vi] induces a forest for each i. It is called linear vertex arboricity if the forests are linear (collection of paths). If G is the line graph of G0, then this equals the (linear) arboricity of G0 [2]. 3.2.4 Average Degree The average degree of a graph G = (V; E) is 2jEj=jV j. 3.3 Graph Intersection Parameters Let G be a class of graphs and let G = (V; E) not necessarily in G. Let p be the T smallest number p of sets Ei with E = i≤p Ei and each (V; Ei) 2 G. Then, p is called G's G-intersection number. 3.3.1 Interval-Intersection (Boxicity) The boxicity is the G-intersection number for G being the class of interval graphs. Exceptionally, each clique has boxicity 0. An equivalent alternative definition is the following. An axis-parallel b- dimensional box is a Cartesian product R1 × R2 × ::: × Rb where Ri (for 3 1 ≤ i ≤ b) is a closed interval of the form [ai; bi] on the real line. For a graph G, its boxicity is the minimum dimension b such that G is representable as the intersection graph of (axis-parallel) boxes in b-dimensional space. 3.3.2 Chordal-Intersection (Chordality) The chordality is the G-intersection number for G being the class of chordal graphs. 3.4 Average Distance n P The average distance of a graph G = (V; E) is 1= 2 · u;v2V d(u; v), where d(u; v) is the length of a shortest path between u and v in G. 3.5 Bandwidth The bandwidth bw of a graph G is the maximum \length" of an edge in a one dimensional layout of G. Formally: ml := min f max fji(u) − i(v)jg : i is injectiveg i:V !N fu;vg2E 3.6 Bisection Width The width of a bipartition of a graph is the number of edges going between the parts. The bisection width of a graph G = (V; E) is the smallest width of a bipartition of G such that the difference of the parts' numbers of vertices is at most one. 3.7 Branchwidth A branch decomposition of a hypergraph H = (V; F) is a tuple (T; τ), where T is a rooted binary tree and where τ is a bijection from the leaves of T to the hyperedges F. The order of an edge e in T is the number of vertices v in H such that there are leaves t1; t2 in different connected components of T n e for which τ(t1); τ(t2) both are incident to v. The width of a branch decomposition is the maximum order of edges in T . The branchwidth of H is the minimum width of branch-decompositions of H. 3.8 Chromatic Number The chromatic number χ of a graph G is the smallest number i such that the the vertices of G can be partitioned into i independent sets. 4 3.9 Cliquewidth Let q be a positive integer. We call (G; λ) a q-labeled graph if G is a graph and λ : V (G) ! f1; 2; : : : ; qg is a mapping. The number λ(v) is called label of a vertex v. We introduce the following operations on labeled graphs: (1) For every i in f1; : : : ; qg, we let •i denote the graph with only one vertex that is labeled by i (a constant operation). (2) For every distinct i and j from f1; 2; : : : ; qg, we define a unary operator 0 0 0 ηi;j such that ηi;j(G; λ) = (G ; λ), where V (G ) = V (G), and E(G ) = E(G) [ fvw : v; w 2 V; λ(v) = i; λ(w) = jg. In other words, the operator adds all edges between label-i vertices and label-j vertices. (3) For every distinct i and j from f1; 2; : : : ; qg, we let ρi!j be the unary 0 0 operator such that ρi!j(G; λ) = (G; λ ), where λ (v) = j if λ(v) = i, and λ0(v) = λ(v) otherwise. The operator only changes the labeling so that the vertices that originally had label i will now have label j. (4) Finally, ⊕ is a binary operation that makes the disjoint union, while keeping the labels of the vertices unchanged. Note explicitly that the union is disjoint in the sense that (G; λ) ⊕ (G; λ) has twice the number of vertices of G. A q-expression is a well-formed expression ' written with these symbols. The q-labeled graph produced by performing these operations in order therefore has a vertex for each occurrence of the constant symbol in '; and this q-labeled graph (and any q-labeled graph isomorphic to it) is called the value val(') of '. If a q-expression ' has value (G; λ), we say that ' is a q-expression of G. The cliquewidth of a graph G, denoted by cwd(G), is the minimum q such that there is a q-expression of G. Cliquewidth has been defined by Courcelle and Olariu [7]. The definition above is inspired by Hlinˇen´yet al. [19]. 3.10 Clique Cover A clique cover of a graph G = (V; E) is a partition P of V such that each part in P induces a clique in G. The minimum clique cover of G is a clique cover with a minimum number of parts. Note that the clique cover number of a graph is exactly the chromatic number of its complement. 3.11 Coloring Number Is one larger than the degeneracy. Introduced by Erd}osand Hajnal [13]. 3.12 Degeneracy The degeneracy of a graph G is the maximum, with respect to all subgraphs G0 of G, of the minimum degree of G0. Equivalent definitions include the minimum 5 outdegree over all acyclic orientations of G and the minimum, over all linear ordering so the vertices, of the maximum, over all vertices v, of the number of neighbors of v that occur later in the ordering. 3.13 Density Also known as average degree. 3.14 Distance to Π The distance to Π of a graph G = (V; E) is the minimum size of a set X ⊆ V such that G[V n X] 2 Π, where Π is a graph class, e.