New bounds on the signed total domination number of graphs

In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turan, we present a sharp lower bound on this parameter for graphs with no complete graph of order r+1 as a subgraph. Also, we prove that n-2(s-s') is an upper bound on the signed total domination number of any tree of order n with s support vertices and s' support vertives of degree two. Moreover, we characterize all trees attainig this bound.Comment: This paper contains 11 pages and one figur

Degree Sequence Index Strategy

We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by which to bound graph invariants by certain indices in the ordered degree sequence. As an illustration of the DSI strategy, we show how it can be used to give new upper and lower bounds on the $k$-independence and the $k$-domination numbers. These include, among other things, a double generalization of the annihilation number, a recently introduced upper bound on the independence number. Next, we use the DSI strategy in conjunction with planarity, to generalize some results of Caro and Roddity about independence number in planar graphs. Lastly, for claw-free and $K_{1,r}$-free graphs, we use DSI to generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester

A Note on Outer-Independent 2-Rainbow Domination in Graphs

Let G be a graph with vertex set V(G) and f:V(G)→{∅,{1},{2},{1,2}} be a function. We say that f is an outer-independent 2-rainbow dominating function on G if the following two conditions hold: (i)V∅={x∈V(G):f(x)=∅} is an independent set of G. (ii)∪u∈N(v)f(u)={1,2} for every vertex v∈V∅. The outer-independent 2-rainbow domination number of G, denoted by γoir2(G), is the minimum weight ω(f)=∑x∈V(G)|f(x)| among all outer-independent 2-rainbow dominating functions f on G. In this note, we obtain new results on the previous domination parameter. Some of our results are tight bounds which improve the well-known bounds β(G)≤γoir2(G)≤2β(G), where β(G) denotes the vertex cover number of G. Finally, we study the outer-independent 2-rainbow domination number of the join, lexicographic, and corona product graphs. In particular, we show that, for these three product graphs, the parameter achieves equality in the lower bound of the previous inequality chain

Further Results on the Total Roman Domination in Graphs

[EN] Let G be a graph without isolated vertices. A function f:V(G)-> {0,1,2} is a total Roman dominating function on G if every vertex v is an element of V(G) for which f(v)=0 is adjacent to at least one vertex u is an element of V(G) such that f(u)=2 , and if the subgraph induced by the set {v is an element of V(G):f(v)>= 1} has no isolated vertices. The total Roman domination number of G, denoted gamma tR(G) , is the minimum weight omega (f)=Sigma v is an element of V(G)f(v) among all total Roman dominating functions f on G. In this article we obtain new tight lower and upper bounds for gamma tR(G) which improve the well-known bounds 2 gamma (G)<= gamma tR(G)<= 3 gamma (G) , where gamma (G) represents the classical domination number. In addition, we characterize the graphs that achieve equality in the previous lower bound and we give necessary conditions for the graphs which satisfy the equality in the upper bound above.Cabrera Martínez, A.; Cabrera García, S.; Carrión García, A. (2020). Further Results on the Total Roman Domination in Graphs. Mathematics. 8(3):1-8. https://doi.org/10.3390/math8030349S1883Henning, M. A. (2009). A survey of selected recent results on total domination in graphs. Discrete Mathematics, 309(1), 32-63. doi:10.1016/j.disc.2007.12.044Henning, M. A., & Yeo, A. (2013). Total Domination in Graphs. Springer Monographs in Mathematics. doi:10.1007/978-1-4614-6525-6Henning, M. A., & Marcon, A. J. (2016). Semitotal Domination in Claw-Free Cubic Graphs. Annals of Combinatorics, 20(4), 799-813. doi:10.1007/s00026-016-0331-zHenning, M. . A., & Marcon, A. J. (2016). Vertices contained in all or in no minimum semitotal dominating set of a tree. Discussiones Mathematicae Graph Theory, 36(1), 71. doi:10.7151/dmgt.1844Henning, M. A., & Pandey, A. (2019). Algorithmic aspects of semitotal domination in graphs. Theoretical Computer Science, 766, 46-57. doi:10.1016/j.tcs.2018.09.019Cockayne, E. J., Dreyer, P. A., Hedetniemi, S. M., & Hedetniemi, S. T. (2004). Roman domination in graphs. Discrete Mathematics, 278(1-3), 11-22. doi:10.1016/j.disc.2003.06.004Stewart, I. (1999). Defend the Roman Empire! Scientific American, 281(6), 136-138. doi:10.1038/scientificamerican1299-136Chambers, E. W., Kinnersley, B., Prince, N., & West, D. B. (2009). Extremal Problems for Roman Domination. SIAM Journal on Discrete Mathematics, 23(3), 1575-1586. doi:10.1137/070699688Favaron, O., Karami, H., Khoeilar, R., & Sheikholeslami, S. M. (2009). On the Roman domination number of a graph. Discrete Mathematics, 309(10), 3447-3451. doi:10.1016/j.disc.2008.09.043Liu, C.-H., & Chang, G. J. (2012). Upper bounds on Roman domination numbers of graphs. Discrete Mathematics, 312(7), 1386-1391. doi:10.1016/j.disc.2011.12.021González, Y., & Rodríguez-Velázquez, J. (2013). Roman domination in Cartesian product graphs and strong product graphs. Applicable Analysis and Discrete Mathematics, 7(2), 262-274. doi:10.2298/aadm130813017gLiu, C.-H., & Chang, G. J. (2012). Roman domination on strongly chordal graphs. Journal of Combinatorial Optimization, 26(3), 608-619. doi:10.1007/s10878-012-9482-yAhangar Abdollahzadeh, H., Henning, M., Samodivkin, V., & Yero, I. (2016). Total Roman domination in graphs. Applicable Analysis and Discrete Mathematics, 10(2), 501-517. doi:10.2298/aadm160802017aAmjadi, J., Sheikholeslami, S. M., & Soroudi, M. (2019). On the total Roman domination in trees. Discussiones Mathematicae Graph Theory, 39(2), 519. doi:10.7151/dmgt.2099Cabrera Martínez, A., Montejano, L. P., & Rodríguez-Velázquez, J. A. (2019). Total Weak Roman Domination in Graphs. Symmetry, 11(6), 831. doi:10.3390/sym1106083

Signed double Roman domination on cubic graphs

The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from $\{\pm{}1,2,3\}$ to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility is given whenever (i) vertices labeled $\pm{}1$ have at least one neighbor with label in $\{2,3\}$; (ii) each vertex labeled $-1$ has one $3$-labeled neighbor or at least two $2$-labeled neighbors; and (iii) the sum of labels over the closed neighborhood of any vertex is positive. The cumulative weight of an optimal labeling is called signed double Roman domination number (SDRDN). In this work, we first consider the problem on general cubic graphs of order $n$ for which we present a sharp $n/2+\Theta(1)$ lower bound for the SDRDN by means of the discharging method. Moreover, we derive a new best upper bound. Observing that we are often able to minimize the SDRDN over the class of cubic graphs of a fixed order, we then study in this context generalized Petersen graphs for independent interest, for which we propose a constraint programming guided proof. We then use these insights to determine the SDRDNs of subcubic $2\times m$ grid graphs, among other results

Distributed Distance-rr Dominating Set on Sparse High-Girth Graphs

The dominating set problem and its generalization, the distance-$r$ dominating set problem, are among the well-studied problems in the sequential settings. In distributed models of computation, unlike for domination, not much is known about distance-r domination. This is actually the case for other important closely-related covering problem, namely, the distance-$r$ independent set problem. By result of Kuhn et al. we know the distributed domination problem is hard on high girth graphs; we study the problem on a slightly restricted subclass of these graphs: graphs of bounded expansion with high girth, i.e. their girth should be at least $4r + 3$. We show that in such graphs, for every constant $r$, a simple greedy CONGEST algorithm provides a constant-factor approximation of the minimum distance-$r$ dominating set problem, in a constant number of rounds. More precisely, our constants are dependent to $r$, not to the size of the graph. This is the first algorithm that shows there are non-trivial constant factor approximations in constant number of rounds for any distance $r$-covering problem in distributed settings. To show the dependency on r is inevitable, we provide an unconditional lower bound showing the same problem is hard already on rings. We also show that our analysis of the algorithm is relatively tight, that is any significant improvement to the approximation factor requires new algorithmic ideas