Properties of θ-super positive graphs

Let the matching polynomial of a graph G be denoted by μ(G, x). A graph G is said to be θ-super positive if μ(G, θ) 6= 0 and μ(G \ v, θ) = 0 for all v ∈ V (G). In particular, G is 0-super positive if and only if G has a perfect matching. While much is known about 0-super positive graphs, almost nothing is known about θsuper positive graphs for θ 6= 0. This motivates us to investigate the structure of θ-super positive graphs in this paper. Though a 0-super positive graph need not contain any cycle, we show that a θ-super positive graph with θ 6= 0 must contain a cycle. We introduce two important types of θ-super positive graphs, namely θelementary and θ-base graphs. One of our main results is that any θ-super positive graph G can be constructed by adding certain type of edges to a disjoint union of θ-base graphs; moreover, these θ-base graphs are uniquely determined by G. We also give a characterization of θ-elementary graphs: a graph G is θ-elementary if and only if the set of all its θ-barrier sets form a partition of V (G). Here, θ-elementary graphs and θ-barrier sets can be regarded as θ-analogue of elementary graphs and Tutte sets in classical matching theory. keywords: matching polynomial, Gallai-Edmonds decomposition, elementary graph, barrier sets, extreme sets the electronic journal of combinatorics 19 (2012), #P37 1