In this article, we use a unified approach to prove several classes of planar graphs are DP-3colorable, which extend the corresponding results on 3-choosability.

A set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The size of a smallest determining set for G is called its determining number, Det(G). A graph G is said to be d-distinguishable if there is a coloring of the vertices with d colors so that only the trivial automorphism preserves the color classes. The smallest d for which G is d-distinguishable is its distinguishing number, Dist(G). If Dist(G) = 2, the cost of 2-distinguishing, ρ(G), is the size of a smallest color class over all 2-distinguishing colorings of G. The Mycielskian, µ(G), of a graph G is constructed by adding a shadow master vertex w, and for each vertex vi of G adding a shadow vertex ui, with edges so that the neighborhood of ui in µ(G) is the same as the neighborhood of vi in G with the addition of w. That is, N (ui) = NG(vi) ∪ {w}. The generalized Mycielskian µ (t) (G) of a graph G is a Mycielskian graph with t layers of shadow vertices, each with edges to layers above and below, and the shadow master only adjacent to the top layer of shadow vertices. A graph is twin-free if it has no pair of vertices with the same set of neighbors. This paper examines the determining number and, when relevant, the cost of 2-distinguishing for Mycielskians and generalized Mycielskians of simple graphs with no isolated vertices. In particular, if G = K2 is twin-free with no isolated vertices, then Det(µ (t) For G with twins, we develop a framework using quotient graphs with respect to equivalence classes of twin vertices to give bounds on the determining number of Mycielskians. Moreover, we identify classes of graphs with twins for which Det(µ (t) (G)) = (t+1) Det(G).

Context: This paper reports on a qualitative study looking at the perceptions of clients who use drugs and are on low income. Research suggests that counselling services can experience difficulties in reaching and retaining such clients, and outcomes of counselling can be disappointing. Aim: To learn from clients who had engaged in counselling for over six months what difference it had made to their lives. Method: Grounded theory methodology was used to analyse semi-structured interviews with six participants. Findings: Findings included changes in the clients' internal world, their connectedness with society, their familiarity with counselling, and their perception of the relationship with the counsellor. Factors within the counselling process which helped and hindered change were identified. The study documents how participants were able to use counselling to improve their lives.

In online list coloring (introduced by Zhu and by Schauz in 2009), on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset to receive that color. The paint number of a graph G is the least k such that there is an algorithm to produce a successful coloring with no vertex being shown more than k times; it is at least the choice number. We study paintability of joins with complete or empty graphs, obtaining a partial result toward the paint analogue of Ohba's Conjecture. We also determine upper and lower bounds on the paint number of complete bipartite graphs and characterize 3-paintcritical graphs.