Weakly Distinguishing Graph Polynomials Johann A. Makowsky Vsevolod Rakita Faculty of Computer Science, Technion : Israel Institute of Technology

Faculty of Mathematics, Technion : Israel Institute of Technology

Definitions Conjecture (Bollobás,Pebody,Riordan) Examples of Graph Polynomials

• A graph polynomial is a function which maps Almost all graphs are determined by their Chromatic Polynomial. • The Chromatic Polynomial: for a graph graphs into a polynomial ring and is invariant Almost all graphs are determined by their Tutte Polynomial. G and n ∈ N, define χ(G, n) to be the number under graph isomorphisms. of proper colourings of G with n colours. It • For a graph polynomial P , we say a graph G is can be shown than χ(G, n) is polynomial in n, P -unique, or determinded by P , if every and is called the chromatic polynomial. We can find sufficient conditions on a graph graph H with P (G; X) = P (H; X) is The Tutte Polynomial: the chromatic By counting the number of possible C polynomials, property A for some interesting graph polynomials • isomorphic to G. polynomial can be shown to be an evaluation we can prove the following theorem: to be weakly distinguishing on A. We use the of a more general bivariate polynomial, called • A statement holds for almost all graphs if the following theorem: proportion of graphs of order n for which it Theorem 1 the Tutte Polynomial defined using the holds, tends to 1, when n tends to infinity. Theorem(McDiarmid,Steger,Welsh) recurrence relation Let C be a graph property that has an indepen- T (G, x, y) = T (G/e, x, y) + T (G\e, x, y) were • A graph H is a P -mate of G if Let C be a non-empty, addable, proper minor dence (clique) function f that satisfies that for all e is an edge in G that is neither a bridge nor a P (G; X) = P (H; X) but H is not isomorphic closed graph property, let H ∈ C be a connected n ∈ N, f(n) ≥ n/a for some fixed a ∈ N. Then loop, G/e is G with e contracted and G\e is to G. graph, and let R be a random graph selected PC is weakly distinguishing. n G with e deleted, with T (G, x, y) = xiyj If G • Let A be a graph property. A graph uniformly at random from the graphs of order n has i bridges, j loops and no other edges. polynomial P is almost complete (on A) if in C. Denote by fH(Rn) the number of pendant • The Characteristic Polynomial of a almost all graphs G (in A) are P -unique, and Theorem 1 applies in the following cases: appearances of H in Rn. Then there are con- graph G is the characteristic polynomial of the it is weakly distinguishing (on A) if almost all (i) Let C be the property containing only edgeless stants α > 0,n0 ∈ N such that for all n > n0, graphs G (in A) have a P -mate. −αn adjacency matrix of G graphs. Clearly C has an independence function P[fH(Rn) ≤ αn] < e • The Domination Polynomial is the • Let G be a graph, C be a class of graphs, and (the identity). Pc is called the Independence denote by Cˆ the class of complement graphs H¯ Polynomial generating function of dominating sets in G, . P |D| By finding graphs H such that if a graph G has H that is Dom(G, x) = D x were the sum is of graphs H ∈ C. The C-polynomial of G is (ii) Let C be the property containing only complete P |A| as a pendent appearance than G has a over all dominating sets in G defined as PC(G; X) = A∈V (G):G[A]∈C X , graphs. Clearly C has an independence function ˆ (Tutte/characteristic/domination) mate, we can and the C-polynomial of G is defined as (the identity). Pc is called the Clique P |A| P ˆ(G; X) = ˆ X . Polynomial prove: C A∈V (G):G[A]∈C Examples of Minor Closed • Let C be a graph property. We say that a (iii) A graph G is k-degenerate if every induced Theorem 2 Addable Properties function f : N → N is an independence subgraph of G has a vertex of degree at most k. It (clique) function for C if for every graph k G Let A be a proper minor closed addable property. is easy to see that every -degenerate graph of • Forests G ∈ C, the graph G has an independent set l n m Then The Tutte, Characteristic and Domination order n has an independent set of size k+1 . polynomials are weakly distinguishing on A • Planar Graphs (clique) of size f(|V (G)|). (iv) Among the k-degenrate graphs we find the graphs • Outerplanar Graphs • A graph property A is said to be Addable if a of tree-width at most k, graphs of degree at graph G is in A if and only if every connected • Graphs with treewidth at most k most k, and planar graphs. The fact that the Characteristic polynomial of a component of G is in A, and if for every graph • k colourable graphs (v)A k-colorable graphs G has an independent set graph is weakly distinguishing on proper minor G ∈ A whenever v and u are vertices in lnm Graphs with no cycles of length greater than k of size at least . closed addable properties is particularly interesting, • different components of G, the graph obtained k as it extends a classic result, proven by Schwenk, • Graphs with no Kk-minors from G by adding an edge between v and u is By theorem 1, if C is any of the properties above, that the characteristic polynomial is weakly • Series-parallel graphs also in A. PC is weakly distinguishing. distinguishing on trees. • A graph property A is said to be minor closed if it is closed under deleting vertices and edges and under contracting edges.