On the Double Roman Domination in Generalized Petersen Graphs P(5k,k)

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f(u)=1 is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G equals the minimum weight of a double Roman dominating function of G. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P(5k,k). It is proven that γdR(P(5k,k))=8k for k≡2,3mod5 and 8k≤γdR(P(5k,k))≤8k+2 for k≡0,1,4mod5. We also improve the upper bounds for generalized Petersen graphs P(20k,k).

On 3-Rainbow Domination Number of Generalized Petersen Graphs P(6k,k)

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for some infinite families, exact values are established; in all other cases, the lower and upper bounds with small gaps are given. We also define singleton rainbow domination, where the sets assigned have a cardinality of, at most, one, and provide analogous results for this special case of rainbow domination.

Hamilton-Connected Mycielski Graphs∗

Jarnicki, Myrvold, Saltzman, and Wagon conjectured that if G is Hamilton-connected and not K 2 , then its Mycielski graph μ G is Hamilton-connected. In this paper, we confirm that the conjecture is true for three families of graphs: the graphs G with δ G > V G / 2 , generalized Petersen graphs G P n , 2 and G P n , 3 , and the cubes G 3 . In addition, if G is pancyclic, then μ G is pancyclic.

Fast Recognition of Some Parametric Graph Families

Recognizing graphs with high level of symmetries is hard in general, and usually requires additional structural understanding. In this paper we study a particular graph parameter and motivate its usage by devising eÿcient recognition algorithm for the family of I-graphs. For integers m a simple graph is cycle regular if every path of length ` belongs to exactly cycles of length m. We identify all cycle regular I-graphs and, as a conse-quence, describe linear recognition algorithm for the observed family. Similar procedure can be used to devise the recog-nition algorithms for Double generalized Petersen graphs and folded cubes. Besides that, we believe the structural observations and methods used in the paper are of independent interest and could be used for solving other algorithmic problems.

On 2-Rainbow Domination Number of Generalized Petersen Graphs P(5k,k)

We obtain new results on 2-rainbow domination number of generalized Petersen graphs P(5k,k). In some cases (for some infinite families), exact values are established, and in all other cases lower and upper bounds are given. In particular, it is shown that, for k>3, γr2(P(5k,k))=4k for k≡2,8mod10, γr2(P(5k,k))=4k+1 for k≡5,9mod10, 4k+1≤γr2(P(5k,k))≤4k+2 for k≡1,6,7mod10, and 4k+1≤γr2(P(5k,k))≤4k+3 for k≡0,3,4mod10.