BACKGROUND In addition to its established role in oncologic imaging, F-fluorodeoxyglucose positron emission tomography/computed tomography (F-FDG PET/CT) is useful for the assessment of inflammatory activity. However, subacute thyroiditis (SAT) in thyrotoxicosis is rarely detected during these scans. CASE A 66-year-old man with SAT in thyrotoxicosis demonstrated symptoms of transient fatigue,...

Given a family of graphs $\mathcal{F}$, we define the $\mathcal{F}$-saturation game as follows. Two players alternate adding edges to an initially empty graph on $n$ vertices, with only constraint being that neither player can add edge creates subgraph in $\mathcal{F}$. The ends when no more be added graph. One wishes end quickly possible, while other prolong game. We let $\textrm{sat}_g(n,\mat...

(k, s)-SAT is the propositional satisfiability problem restricted to instances where each clause has exactly k distinct literals and every variable occurs at most s times. It is known that there exists an exponential function f such that for s ≤ f(k) all (k, s)-SAT instances are satisfiable, but (k, f(k)+1)-SAT is already NP-complete (k ≥ 3). Exact values of f are only known for k = 3 and k = 4...

Let F be a family of graphs. A graph G is F-saturated if G contains no member of F as a subgraph but G + e contains some member of F whenever e ∈ E(G). The saturation number and extremal number of F , denoted sat(n,F) and ex(n,F) respectively, are the minimum and maximum numbers of edges among n-vertex Fsaturated graphs. For k ∈ N, let Fk and F ′ k be the families of k-connected and k-edgeconne...

In the maximum 2-satisfiability problem (abbreviated as Max 2-Sat), one is given a Boolean formula in conjunctive normal form, such that each clause contains at most two literals. The task is to find an assignment to the variables of the formula such that a maximum number of clauses is satisfied. Max 2-Sat is a classic optimization problem. Its decision version was proved NP-complete by Garey, ...

The representation of the set of falsifying assignments of clauses via binary patterns has been useful in the design of algorithms for solving #FAL (counting the number of falsifying assignments of conjunctive forms (CF)). Given as input a CF formula F expressed by m clauses defined over n variables, we present a deterministic algorithm for computing #SAT (F ). Initially, our algorithm computes...

Consider SAT, the prototypical example of an NP-complete problem. An instance of this problem consists of a Boolean function f (x1, . . . ,xn) = c1 ∧ . . .∧ cm; the SAT problem asks you to determine whether there exists a satisfying assignment—that is, an input (a1, . . . ,an) such that f (a1, . . . ,an) = 1. UNIQUE-SAT is a variant of SAT that poses the same problem with the restriction that f...

Given graphs H and F , a subgraph G ⊆ H is an F -saturated subgraph of H if F * G, but F ⊆ G + e for all e ∈ E(H) \ E(G). The saturation number of F in H, denoted sat(H,F ), is the minimum number of edges in an F -saturated subgraph of H. In this paper we study saturation numbers of tripartite graphs in tripartite graphs. For ` ≥ 1 and n1, n2, and n3 sufficiently large, we determine sat(Kn1,n2,...

For a family F of graphs, a graph G is F-saturated if G contains no member of F as a subgraph, but for any edge uv in G, G+ uv contains some member of F as a subgraph. The minimum number of edges in an F-saturated graph of order n is denoted sat(n,F). A subdivision of a graph H, or an H-subdivision, is a graph G obtained from H by replacing the edges of H with internally disjoint paths of arbit...

We consider the ~ATISFIABILITY problem (~-SAT): Given a family F of n clauses cl, ._, , c, in conjunctive normal form, each consisting of k literals corresponding to k different variables of a set of r 2 k 2 1 boolean variables, is F satisfiable? By k-SAT@ no) we denote the k-SAT problem restricted to families with n > n,(r) clauses. We prove that for each k > 3 and each integer I > 4 such that...