Positive Semidefinite Maximum Nullity and Positive Semidefinite Zero Forcing Number of Multigraphs with Tree-width at most 2

Zero forcing is a process of changing the color of vertices on a graph G from white to black following a color change rule. The zero forcing number, Z(G), is the minimum number of initially black vertices needed to obtain an all black final coloring. A real symmetric matrix A=[a_ij] is described by G if, for i≠j, a_ij≠0 whenever i and j are adjacent in G and a_ij=0 whenever {i,j} is not an edge in G. The maximum positive semidefinite nullity of G, M₊(G), is the maximum nullity over all matrices described by G. Zero forcing arose both in the study of the maximum nullity of a family of symmetric matrices described by a graph (because M(G)≤Z(G) for all G) and in the control of quantum systems in physics. The maximum positive semidefinite nullity of G, M₊(G), is the maximum nullity over all positive semidefinite matrices described by G. The positive semidefinite zero forcing number of G, Z₊(G), is the minimum number of initally black vertices needed to obtain an all black final coloring using the positive semidefinite color change rule and has the property M₊(G)≤Z₊(G). We show that for graphs having tree width at most 2, M₊(G)=Z₊(G) and relate this to other graph parameters. We use multigraphs and define positive semidefinite zero forcing for multigraphs.