The domination polynomial D(G, x) of a graph G is the generating function of its dominating sets. We prove that D(G, x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G, x) for arbitrary graphs and for various special cases. We give splitting formulas for D(G, x) based on articulation vertices, and more generally, on splitting sets of vertices.where G i are obtained from G using various vertex and edge elimination operations and the g i (x)'s are given rational functions. For example, it is well-known that the independence polynomial satisfies a linear recurrence relation with respect to two vertex elimination operations, the deletion of a vertex and the deletion of vertex's closed neighborhood. Other prominent graph polynomials in the literature satisfy similar recurrence relations with respect to vertex and edge elimination operations, among them the matching polynomial, the chromatic polynomial and the vertex-cover polynomial, see e.g. [11].In contrast, it is significantly harder to find recurrence relations for the domination polynomial. We show in Theorem 2.4 that D(G, x) does not satisfy any linear recurrence relation which applies only the commonly used vertex operations of deletion, extraction, contraction and neighborhood-contraction. Nor does D(G, x) satisfy any linear recurrence relation using only edge deletion, contraction and extraction.In spite of this non-existence result, we give in this paper an abundance of recurrence relations and splitting formulas for the domination polynomial.The domination polynomial was studied recently by several authors, see [1,2,3,5,6,7,8,9,10,12]. The previous research focused mainly on the roots of domination polynomials and on the domination polynomials of various classes of special graphs. In [12] it is shown that computing the domination polynomial D(G, x) of a graph G is NPhard and some examples for graphs for which D(G, x) can be computed efficiently are given. Some of our results, e.g. Theorem 5.14, lead to efficient schemes to compute the domination polynomial.An outline of the paper is as follows. In Section 2 we give a recurrence relation for arbitrary graphs. In Section 3 we give simple recurrence steps in special cases, which allow us e.g. to dispose of triangles, induced 5-paths and irrelevant edges. In Section 4 we consider graphs of connectivity 1, and give several splitting formulas for them. In Section 5 we generalize the results of the previous section to arbitrary separating vertex sets. In Section 6 we show a recurrence relation for arbitrary graphs which uses derivatives of domination polynomials.

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of the type A⊆S f (A) where S is a finite set and f is a mapping from the power set of S to an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the Möbius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical Möbius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).

Whitney's Broken-cycle Theorem states the chromatic polynomial of a graph as a sum over special edge subsets. We give a definition of cycles in hypergraphs that preserves the statement of the theorem there. * Supported by European Social Fond grant 080940498.

Let G = (V, E ) be a graph and σ (G) the number of independent (vertex) sets of G. Then the Merrifield-Simmons conjecture states that the sign of the term σonly depends on the parity of the distance of the vertices u, v ∈ V in G. We prove that the conjecture holds for bipartite graphs by considering a generalization of the term, where vertex subsets instead of vertices are deleted.

Motivated by the definition of the edge elimination polynomial of a graph we define the covered components polynomial counting spanning subgraphs with respect to their number of components, edges and covered components. We prove a recurrence relation, which shows that both graph polynomials are substitution instances of each other. We give some properties of the covered components polynomial and some results concerning relations to other graph polynomials. * trinks@hs-mittweida.