8 research outputs found
Pebbling in Semi-2-Trees
Graph pebbling is a network model for transporting discrete resources that
are consumed in transit. Deciding whether a given configuration on a particular
graph can reach a specified target is -complete, even for diameter
two graphs, and deciding whether the pebbling number has a prescribed upper
bound is -complete. Recently we proved that the pebbling number
of a split graph can be computed in polynomial time. This paper advances the
program of finding other polynomial classes, moving away from the large tree
width, small diameter case (such as split graphs) to small tree width, large
diameter, continuing an investigation on the important subfamily of chordal
graphs called -trees. In particular, we provide a formula, that can be
calculated in polynomial time, for the pebbling number of any semi-2-tree,
falling shy of the result for the full class of 2-trees.Comment: Revised numerous arguments for clarity and added technical lemmas to
support proof of main theorem bette
Critical Pebbling Numbers of Graphs
We define three new pebbling parameters of a connected graph , the -,
-, and -critical pebbling numbers. Together with the pebbling number, the
optimal pebbling number, the number of vertices and the diameter of the
graph, this yields 7 graph parameters. We determine the relationships between
these parameters. We investigate properties of the -critical pebbling
number, and distinguish between greedy graphs, thrifty graphs, and graphs for
which the -critical pebbling number is .Comment: 26 page
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
The Complexity Of Pebbling Reachability And Solvability In Planar And Outerplanar Graphs
Given a simple, connected graph, a pebbling configuration is a function from its vertex set to the nonnegative integers. A pebbling move between adjacent vertices removes two pebbles from one vertex and adds one pebble to the other. A vertex r is said to be reachable from a configuration if there exists a sequence of pebbling moves that places at least one pebble on r. A configuration is solvable if every vertex is reachable. We prove that determining reachability of a vertex and solvability of a configuration are NP-complete on planar graphs. We also prove that both reachability and solvability can be determined in O(n6)time on planar graphs with diameter two. Finally, for outerplanar graphs, we present a linear algorithm for determining reachability and a quadratic algorithm for determining solvability. To prove this result, we provide linear algorithms to determine all possible maximal configurations of pebbles that can be placed on the endpoints of a path and on two adjacent vertices in a cycle