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In this note we consider the following problem: Given a graph [Formula: see text] and a subgraph [Formula: see text], what is the smallest subset [Formula: see text] of edges in [Formula: see text] that needs to be deleted from the graph to make it [Formula: see text]-free? Several algorithmic results are presented. First, using the general framework of Courcelle [9], we show that, for a fixed subgraph [Formula: see text], the problem can be solved in linear time on graphs of bounded treewidth. It is known that the constant hidden in the big-O notation of Courcelle algorithm is big which makes the approach impractical. Thus, we present two explicit linear time dynamic programming algorithms on graphs of bounded treewidth for restricted settings of the problem with reasonable constants. Third, using the linear time algorithm for graphs of bounded treewidth, we design a Baker's type polynomial time approximation scheme for the problem on planar graphs.

This paper addresses the problem of finding time-optimal trajectories in two-dimensional space, where the mover’s speed monotonically decreases or increases in one of the space’s coordinates. We address such problems in different settings for the velocity function and in the presence of obstacles. First, we consider the problem without any obstacles. We show that the problem with linear speed decrease is reducible to the de l’hôpital’s problem, and that, for this case, the time-optimal trajectory is a circular segment. Next, we show that the problem with linear speed decreases and rectilinear obstacles can be solved in polynomial time by a dynamic program. Finally, we consider the case without obstacles, where the medium is non-uniform and the mover’s velocity is a piecewise linear function. We reduce this problem to that of solving a system of polynomial equations of fixed degree, for which algebraic elimination theory allows us to solve the problem to optimality.

Given a graph G with tree-width @w(G), branch-width @b(G), and side size of the largest square grid-minor @q(G), it is known that @q(G)@[email protected](G)@[email protected](G)[email protected][email protected](G). In this paper, we introduce another approach to bound the side size of the largest square grid-minor specifically for planar graphs. The approach is based on measuring the distances between the faces in an embedding of a planar graph. We analyze the tightness of all derived bounds. In particular, we present a class of planar graphs where @q(G)[email protected](G)

Given a simple planar graph with tree-width w and side size of the largest square grid minor g, it is known that g ⩽ w ⩽ 5 g − 1 . Thus, the side size of the largest grid minor is a constant approximation for the tree-width in planar graphs. In this work we analyze the lower bounds of this approximation. In particular, we present a class of planar graphs with ⌊ 3 g / 2 ⌋ − 1 ⩽ w ⩽ ⌊ 3 g / 2 ⌋ . We conjecture that in the worst case w = 2 g + o ( g ) . For this conjecture we have two candidate classes of planar graphs.