Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of 2 pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of pebbles t so that, from any initial configuration of t pebbles on its vertices, one can place a pebble on any given target vertex via such pebbling steps. It is known that deciding whether a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter 2 graphs, and that deciding whether the pebbling number has a prescribed upper bound is Π[P over 2]-complete. On the other hand, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter 2 graphs, and more. Moreover, graphs having minimum pebbling number are called Class 0, and many authors have studied which graphs are Class 0 and what graph properties guarantee it, with no characterization in sight. In this paper we investigate an important family of diameter 3 chordal graphs called split graphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide a formula for the pebbling number of a split graph, along with an algorithm for calculating it that runs in O(n[superscript β]) time, where β = 2ω/(ω + 1) [= over ∼] 1.41 and ω [= over ∼] 2.376 is the exponent of matrix multiplication. Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.

Previous studies have shown that parental protectiveness is associated with increased pain and disability in Functional Abdominal Pain Disorder (FAPD) but the role that perceived child self-efficacy may play remains unclear. One reason why parents may react protectively towards their child’s pain is that they perceive their child to be unable to cope or function normally while in pain (perceived low self-efficacy). This study sought to examine (a) the association between parent-perceived child pain self-efficacy and child health outcomes (symptom severity and disability); and (b) the role of parental protectiveness as a mediator of this association. Participants were 316 parents of children aged 7–12 years with FAPD. Parents completed measures of perceived child self-efficacy when in pain, their own protective responses to their child’s pain, child gastrointestinal (GI) symptom severity, and child functional disability. Parent-perceived child self-efficacy was inversely associated with parent-reported child GI symptom severity and disability, and parental protectiveness mediated these associations. These results suggest that parents who perceive their child to have low self-efficacy to cope with pain respond more protectively when they believe he/she is in pain, and this, in turn, is associated with higher levels of GI symptoms and disability in their child. This finding suggests that directly addressing parent beliefs about their child’s ability to manage pain should be included as a component of FAPD, and potentially other child treatment interventions.