arXiv subjects

Randy Davila

Publications and source records attributed to Randy Davila.

35 records · Page 2Linked to original sources

arXiv2019

New results relating independence and matchings

In this paper we study relationships between the \emph{matching number}, written $μ(G)$, and the \emph{independence number}, written $α(G)$. Our first main result is to show \[ α(G) \le μ(G) + |X| - μ(G[N_G[X]]), \] where $X$ is \emph{any} intersection of maximum independent sets in $G$. Our second main result is to show \[ δ(G)α(G) \le Δ(G)μ(G), \] where $δ(G)$ and $Δ(G)$ denote the minimum and maximum vertex degrees of $G$, respectively. These results improve on and generalize known relations between $μ(G)$ and $α(G)$. Further, we also give examples showing these improvements.

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arXiv2018

A lower bound on the zero forcing number

In this note, we study a dynamic vertex coloring for a graph $G$. In particular, one starts with a certain set of vertices black, and all other vertices white. Then, at each time step, a black vertex with exactly one white neighbor forces its white neighbor to become black. The initial set of black vertices is called a \emph{zero forcing set} if by iterating this process, all of the vertices in $G$ become black. The \emph{zero forcing number} of $G$ is the minimum cardinality of a zero forcing set in $G$, and is denoted by $Z(G)$. Davila and Kenter have conjectured in 2015 that $Z(G)\geq (g-3)(δ-2)+δ$ where $g$ and $δ$ denote the girth and the minimum degree of $G$, respectively. This conjecture has been proven for graphs with girth $g \leq 10$. In this note, we present a proof for $g \geq 5$, $δ\geq 2$, thereby settling the conjecture.

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arXiv2018

A Vizing-type result for semi-total domination

A set of vertices $S$ in a simple isolate-free graph $G$ is a semi-total dominating set of $G$ if it is a dominating set of $G$ and every vertex of $S$ is within distance 2 or less with another vertex of $S$. The semi-total domination number of $G$, denoted by $γ_{t2}(G)$, is the minimum cardinality of a semi-total dominating set of $G$. In this paper, we study semi-total domination of Cartesian products of graphs. Our main result establishes that for any graphs $G$ and $H$, $γ_{t2}(G\,\square\, H)\ge \frac{1}{3}γ_{t2}(G)γ_{t2}(H)$.

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arXiv2018

Zero Forcing in Claw-Free Cubic Graphs

The zero forcing number of a simple graph, written $Z(G)$, is a NP-hard graph invariant which is the result of the zero forcing color change rule. This graph invariant has been heavily studied by linear algebraists, physicists, and graph theorist. It's broad applicability and interesting combinatorial properties have attracted the attention of many researchers. Of particular interest, is that of bounding the zero forcing number from above. In this paper we show a surprising relation between the zero forcing number of a graph and the independence number of a graph, denoted $α(G)$. Our main theorem states that if $G \ne K_4$ is a connected, cubic, claw-free graph, then $Z(G) \le α(G) + 1$. This improves on best known upper bounds for $Z(G)$, as well as known lower bounds on $α(G)$. As a consequence of this result, if $G \ne K_4$ is a connected, cubic, claw-free graph with order $n$, then $Z(G) \le \frac{2}{5}n + 1$. Additionally, under the hypothesis of our main theorem, we further show $Z(G) \le α'(G)$, where $α'(G)$ denotes the matching number of $G$.

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arXiv2018

Matching, Path Covers, and Total Forcing Sets

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. If the initial set $S$ has the added property that it induces a subgraph of $G$ without isolated vertices, then $S$ is called a total forcing set in $G$. The minimum cardinality of a total forcing set in $G$ is its total forcing number, denoted $F_t(G)$. The path cover number of $G$, denoted $\pc(G)$, is the minimum number of vertex disjoint paths such that every vertex belongs to a path in the cover, while the matching number of $G$, denoted $α'(T)$, is the number of edges in a maximum matching of $G$. Let $T$ be a tree of order at least two. We observe that $\pc(T) + 1 \le F_t(T) \le 2\pc(T)$, and we prove that $F_t(T) \le α'(T) + \pc(T)$. Further, we characterize the extremal trees achieving equality in these bounds.

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arXiv2017

Extremal $k$-forcing sets in oriented graphs

This article studies the \emph{$k$-forcing number} for oriented graphs, generalizing both the \emph{zero forcing number} for directed graphs and the $k$-forcing number for simple graphs. In particular, given a simple graph $G$, we introduce the maximum (minimum) oriented $k$-forcing number, denoted $\MOF_k(G)$ ($\mof_k(G)$), which is the largest (smallest) $k$-forcing number among all possible orientations of $G$. These new ideas are compared to known graph invariants and it is shown that, among other things, $\mof(G)$ equals the path covering number of $G$ while $\MOF_k(G)$ is greater than or equal to the independence number of $G$ -- with equality holding if $G$ is a tree or if $k$ is at least the maximum degree of $G$. Along the way, we also show that many recent results about $k$-forcing number can be modified for oriented graphs.

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arXiv2017

Total Forcing and Zero Forcing in Claw-Free Cubic Graphs

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set (zero forcing set) of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. If the initial set $S$ has the added property that it induces a subgraph of $G$ without isolated vertices, then $S$ is called a total forcing set in $G$. The total forcing number of $G$, denoted $F_t(G)$, is the minimum cardinality of a total forcing set in $G$. We prove that if $G$ is a connected cubic graph of order~$n$ that has a spanning $2$-factor consisting of triangles, then $F_t(G) \le \frac{1}{2}n$. More generally, we prove that if $G$ is a connected, claw-free, cubic graph of order~$n \ge 6$, then $F_t(G) \le \frac{1}{2}n$, where a claw-free graph is a graph that does not contain $K_{1,3}$ as an induced subgraph. The graphs achieving equality in these bounds are characterized.

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arXiv2017

On the Total Forcing Number of a Graph

Let $G$ be a simple and finite graph without isolated vertices. In this paper we study forcing sets (zero forcing sets) which induce a subgraph of $G$ without isolated vertices. Such a set is called a total forcing set, introduced and first studied by Davila \cite{Davila}. The minimum cardinality of a total forcing set in $G$ is the total forcing number of $G$, denoted $F_t(G)$. We study basic properties of $F_t(G)$, relate $F_t(G)$ to various domination parameters, and establish $NP$-completeness of the associated decision problem for $F_t(G)$. We also prove that if $G$ is a connected graph of order $n \ge 3$ and maximum degree $Δ$, then $F_t(G) \le ( \fracΔ{Δ+1} ) n$, with equality if and only if $G$ is a complete graph $K_{Δ+ 1}$.

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arXiv2017

Total Forcing Sets in Trees

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. If the initial set $S$ has the added property that it induces a subgraph of $G$ without isolated vertices, then $S$ is called a total forcing set in $G$. The minimum cardinality of a total forcing set in $G$ is its total forcing number, denoted $F_t(G)$. We prove that if $T$ is a tree of order $n \ge 3$ with maximum degree~$Δ$, then $F_t(T) \le \frac{1}Δ((Δ- 1)n + 1)$, and we characterize the infinite family of trees achieving equality in this bound. We also prove that if $T$ is a non-trivial tree with $n_1$ leaves, then $F_t(T) \ge n_1$, and we characterize the infinite family of trees achieving equality in this bound. As a consequence of this result, the total forcing number of a non-trivial tree is strictly greater than its forcing number. In particular, we prove that if $T$ is a non-trivial tree, then $F_t(T) \ge F(T)+1$, and we characterize extremal trees achieving this bound.

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arXiv2017

A note on sub-total domination in graphs

Let $G$ be a simple and finite graph without isolated vertices. In this paper we introduce and study a new degree sequence derived invariant called the \emph{sub-total domination number}, denoted $\sub_t(G)$. In particular, we show that $\sub_t(G)$ serves as a lower bound on $γ_t(G)$, where $γ_t(G)$ denotes the heavily studied \emph{total domination number} of $G$.

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arXiv2016

The sub-$k$-domination number of a graph with applications to $k$-domination

In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph $G$, called the sub-$k$-domination number and denoted $sub_k(G)$. We show that $sub_k(G)$ is a computationally efficient sharp lower bound on the $k$-domination number of $G$, and improves on several known lower bounds. We also characterize the sub-$k$-domination numbers of several families of graphs, provide structural results on sub-$k$-domination, and explore properties of graphs which are $sub_k(G)$-critical with respect to addition and deletion of vertices and edges.

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arXiv2016

The forcing number of graphs with a given girth

In this paper, we study a dynamic coloring of the vertices of a graph $G$ that starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. The forcing number, originally known as the \emph{zero forcing number}, and denoted $F(G)$, of $G$ is the cardinality of a smallest forcing set of $G$. We study lower bounds on the forcing number in terms of its minimum degree and girth, where the girth $g$ of a graph is the length of a shortest cycle in the graph. Let $G$ be a graph with minimum degree $δ\ge 2$ and girth~$g \ge 3$. Davila and Kenter [Theory and Applications of Graphs, Volume 2, Issue 2, Article 1, 2015] conjecture that $F(G) \ge δ+ (δ-2)(g-3)$. This conjecture has recently been proven for $g \le 6$. The conjecture is also proven when the girth $g \ge 7$ and the minimum degree is sufficiently large. In particular, it holds when $g = 7$ and $δ\ge 481$, when $g = 8$ and $δ\ge 649$, when $g = 9$ and $δ\ge 30$, and when $g = 10$ and $δ\ge 34$. In this paper, we prove the conjecture for $g \in \{7,8,9,10\}$ and for all values of $δ\ge 2$.

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arXiv2016

Bounds on the connected forcing number of a graph

In this paper, we study (zero) forcing sets which induce connected subgraphs of a graph. The minimum cardinality of such a set is called the connected forcing number of the graph. We provide sharp upper and lower bounds on the connected forcing number in terms of the minimum degree, maximum degree, girth, and order of the graph.

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arXiv2016

Characterizations of the Connected Forcing Number of a Graph

Zero forcing is a dynamic graph coloring process whereby a colored vertex with a single uncolored neighbor forces that neighbor to be colored. This forcing process has been used to approximate certain linear algebraic parameters, as well as to model the spread of diseases and information in social networks. In this paper, we introduce and study the connected forcing process -- a restriction of zero forcing in which the initially colored set of vertices induces a connected subgraph. We show that the connected forcing number -- the cardinality of the smallest initially colored vertex set which forces the entire graph to be colored -- is a sharp upper bound to the maximum nullity, path cover number, and leaf number of the graph. We also give closed formulas and bounds for the connected forcing numbers of several families of graphs including trees, hypercubes, and flower snarks, and characterize graphs with extremal connected forcing numbers.

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arXiv2015

Lower Bounds on the Distance Domination Number of a Graph

For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number, $γ_k(G)$, of $G$ is the minimum cardinality of a $k$-dominating set of $G$. In this paper, we establish lower bounds on the $k$-domination number of a graph in terms of its diameter, radius and girth. We prove that for connected graphs $G$ and $H$, $γ_k(G \times H) \ge γ_k(G) + γ_k(H) -1$, where $G \times H$ denotes the direct product of $G$ and $H$.

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arXiv2014

Bounds for the Zero-Forcing Number of Graphs with Large Girth

We investigate the zero-forcing number for triangle-free graphs. We improve upon the trivial bound, $δ\le Z(G)$ where $δ$ is the minimum degree, in the triangle-free case. In particular, we show that $2 δ- 2 \le Z(G)$ for graphs with girth of at least 5, and this can be further improved when $G$ has a small cut set. Using these results, we are able to prove the Graph Complement Conjecture on minimum rank for a large class of graphs. Lastly, we make a conjecture that the lower bound for $Z(G)$ increases as a function of the girth, $g$, and $δ$.

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arXiv2014

Upper bounds on the k-forcing number of a graph

Given a simple undirected graph $G$ and a positive integer $k$, the $k$-forcing number of $G$, denoted $F_k(G)$, is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process described by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored vertex has at most $k$ non-colored neighbors, then each of its non-colored neighbors becomes colored. When $k=1$, this is equivalent to the zero forcing number, usually denoted with $Z(G)$, a recently introduced invariant that gives an upper bound on the maximum nullity of a graph. In this paper, we give several upper bounds on the $k$-forcing number. Notable among these, we show that if $G$ is a graph with order $n \ge 2$ and maximum degree $Δ\ge k$, then $F_k(G) \le \frac{(Δ-k+1)n}{Δ- k + 1 +\min{\{δ,k\}}}$. This simplifies to, for the zero forcing number case of $k=1$, $Z(G)=F_1(G) \le \frac{Δn}{Δ+1}$. Moreover, when $Δ\ge 2$ and the graph is $k$-connected, we prove that $F_k(G) \leq \frac{(Δ-2)n+2}{Δ+k-2}$, which is an improvement when $k\leq 2$, and specializes to, for the zero forcing number case, $Z(G)= F_1(G) \le \frac{(Δ-2)n+2}{Δ-1}$. These results resolve a problem posed by Meyer about regular bipartite circulant graphs. Finally, we present a relationship between the $k$-forcing number and the connected $k$-domination number. As a corollary, we find that the sum of the zero forcing number and connected domination number is at most the order for connected graphs.

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