3-Factor-criticality in double domination edge critical graphs

A vertex subset $S$ of a graph $G$ is a double dominating set of $G$ if $|N[v]\cap S|\geq 2$ for each vertex $v$ of $G$, where $N[v]$ is the set of the vertex $v$ and vertices adjacent to $v$. The double domination number of $G$, denoted by $\gamma_{\times 2}(G)$, is the cardinality of a smallest double dominating set of $G$. A graph $G$ is said to be double domination edge critical if $\gamma_{\times 2}(G+e)<\gamma_{\times 2}(G)$ for any edge $e \notin E$. A double domination edge critical graph $G$ with $\gamma_{\times 2}(G)=k$ is called $k$-$\gamma_{\times 2}(G)$-critical. A graph $G$ is $r$-factor-critical if $G-S$ has a perfect matching for each set $S$ of $r$ vertices in $G$. In this paper we show that $G$ is 3-factor-critical if $G$ is a 3-connected claw-free $4$-$\gamma_{\times 2}(G)$-critical graph of odd order with minimum degree at least 4 except a family of graphs.Comment: 14 page

On Minimum Maximal Distance-k Matchings

We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge analogue of $k$-equipackable graphs. We prove that the recognition of $k$-equimatchable graphs is co-NP-complete for any fixed $k \ge 2$. We provide a simple characterization for the class of strongly chordal graphs with equal $k$-packing and $k$-domination numbers. We also prove that for any fixed integer $\ell \ge 1$ the problem of finding a minimum weight maximal distance-$2\ell$ matching and the problem of finding a minimum weight $(2 \ell - 1)$-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of $\delta \ln |V(G)|$ unless $\mathrm{P} = \mathrm{NP}$, where $\delta$ is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure

An improvement in the two-packing bound related to Vizing\u27s conjecture

Vizing\u27s conjecture states that the domination number of the Cartesian product of graphs is at least the product of the domination numbers of the two factor graphs. In this note we improve the recent bound of Breŝar by applying a technique of Zerbib to show that for any graphs G and H, γ(G x H)≥ γ (G) 2/3(γ(H)-ρ(H)+1), where γ is the domination number, ρ is the 2-packing number, and x is the Cartesian product

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

Total Roman Domination Number of Rooted Product Graphs

[EN] Let G be a graph with no isolated vertex and f:V(G)->{0,1,2} a function. If f satisfies that every vertex in the set {v is an element of V(G):f(v)=0} is adjacent to at least one vertex in the set {v is an element of V(G):f(v)=2}, and if the subgraph induced by the set {v is an element of V(G):f(v)>= 1} has no isolated vertex, then we say that f is a total Roman dominating function on G. The minimum weight omega(f)= n-ary sumation v is an element of V(G)f(v) among all total Roman dominating functions f on G is the total Roman domination number of G. In this article we study this parameter for the rooted product graphs. Specifically, we obtain closed formulas and tight bounds for the total Roman domination number of rooted product graphs in terms of domination invariants of the factor graphs involved in this product.Cabrera Martinez, A.; Cabrera García, S.; Carrión García, A.; Hernandez Mira, FA. (2020). Total Roman Domination Number of Rooted Product Graphs. Mathematics. 8(10):1-13. https://doi.org/10.3390/math8101850S11381