The rubbling number of certain types of Jahangir graphs and t-rubbling number of lamp graphs

Graph rubbling and graph pebbling, known as graph moves in graph theory, are originated from number theory. Graph rubbling is an extension of graph pebbling. In graph pebbling, two pebbles from the vertices are removed, and one pebble from adjacent vertices will be added. Pebbles are thought of as fuel containers in graph theory. In graph pebbling, a vertex is said to be reachable if a pebble can be moved to that vertex using pebble moves. This paper addresses the extension of pebbling, which is called rubbling. Graph rubbling differs from Graph pebbling in the sense that graph rubbling possesses an additional step of adding a pebble on the vertices after the removal of a single pebble, each from the two adjacent vertices. If the adjacent vertices are similar, the rubbling becomes pebbling; however, if they are different, rubbling is called strict rubbling. The t–rubbling number of a graph G is the minimum number of n pebbles in a pebble distribution on the vertices of G with which we can make t–pebbles on any vertices with making them reachable. In this paper, we have extended the t pebbling, which is derived to find the t-rubbling number of Jahangir graphs and lamp graph. We derived the t-rubbling number for Jahangir graph for n = 3, n = 7, and n > 7, whereas for Lamp graph for n = 3, n = 4, n = 5, n = 6, n = 7, and n > 7.