Let G = ( V , E ) be a directed graph and ℓ : V → [ k ] : = { 1 , 2 , … , k } a level assignment such that ℓ ( u ) < ℓ ( v ) for all directed edges ( u , v ) ∈ E . A level-planar drawing of G maps each vertex v to a unique point on the horizontal line ℓ with y -coordinate ℓ ( v ) and each directed edge to a y -monotone Jordan arc between its endpoints such that no two arcs cross in their interior. In the problem Constrained Level Planarity ( CLP for short), we are further given a partial ordering ◁ i of V i : = ℓ − 1 ( i ) for each i ∈ [ k ] , and we seek a level-planar drawing where the linear order ≺ i of the vertices on ℓ i is a linear extension of ◁ i . A special case of this is the problem Partial Level Planarity ( PLP for short), where we are asked to extend a given level-planar drawing H of a subgraph H of G to a complete drawing G of G without modifying the given drawing, i.e., the restriction of G to H must coincide with H . We give a simple polynomial-time algorithm with running time O ( n 5 ) for CLP of single-source graphs that is based on a simplified version of an existing level-planarity testing algorithm for single-source graphs. We introduce a modified type of PQ-tree data structure that is capable of efficiently handling the arising constraints to improve the running time to O ( n + k s ) , where s denotes the size of the constraints. We complement this result by showing that PLP is NP -complete even in very restricted cases. In particular, PLP remains NP -complete even when G has a constant number of levels, and when G is a subdivision of a triconnected planar graph with bounded degree.

We introduce and study a novel majority-based opinion diffusion model. Consider a graph G , which represents a social network. Assume that initially a subset of nodes, called seed nodes or early adopters, are colored either black or white, which correspond to positive or negative opinion regarding a consumer product or a technological innovation. Then, in each round an uncolored node, which is adjacent to at least one colored node, chooses the most frequent color among its neighbors. Consider a marketing campaign which advertises a product of poor quality and its ultimate goal is that more than half of the population believe in the quality of the product at the end of the opinion diffusion process. We focus on three types of attackers which can select the seed nodes in a deterministic or random fashion and manipulate almost half of them to adopt a positive opinion toward the product (that is, to choose black color). We say that an attacker succeeds if a majority of nodes are black at the end of the process. Our main purpose is to characterize classes of graphs where an attacker cannot succeed. In particular, we prove that if the maximum degree of the underlying graph is not too large or if it has strong expansion properties, then it is fairly resilient to such attacks. Furthermore, we prove tight bounds on the stabilization time of the process (that is, the number of rounds it needs to end) in both settings of choosing the seed nodes deterministically and randomly. We also provide several hardness results for some optimization problems regarding stabilization time and choice of seed nodes.