We prove that the \emph{standard zero forcing number} $Z(G)$ and the \emph{positive semidefinite zero forcing number} $Z_+(G)$ are equal for all claw-free graphs $G$. This result resolves a conjecture proposed by the computer program \emph{TxGraffiti} and highlights a connection between these graph invariants in claw-free structures. As a corollary, we show that a graph $G$ is claw-free if and only if every induced subgraph $H \subseteq G$ satisfies $Z(H) = Z_+(H)$.
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…
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 theo…
Given a simple, finite graph with vertex set $V(G)$, we define a zero forcing set of $G$ as follows. Choose $S\subseteq V(G)$ and color all vertices of $S$ blue and all vertices in $V(G) - S$ white. The color change rule is if $w$ is the only white neighbor of blue vertex $v$, then we change the color of $w$ from white to blue. If after applying the color change rule as many times as possible eventually every vertex of $G$ is blue, we call $S$ a zero forcing set of $G$. $Z(G)$ denotes the minimum cardinality of a zero forcing set. Davila and Henning proved in \cite{zerocubic} that for any claw…
In a series of papers, of which this is the first, we study sufficient conditions for Hamiltonicity in terms of forbidden induced subgraphs and extend such results to locally finite infinite graphs. For this we use topological circles within the Freudenthal compactification of a locally finite graph as infinite cycles. In this paper we focus on conditions involving claws, nets and bulls as induced subgraphs. We extend Hamiltonicity results for finite claw-free and net-free graphs by Shepherd to locally finite graphs. Moreover, we generalise a classification of finite claw-free and net-free gra…
In this paper, we showcase the process of using an automated conjecturing program called \emph{TxGraffiti} written and maintained by the second author. We begin by proving a conjecture formulated by \emph{TxGraffiti} that for a claw-free graph $G$, the vertex cover number $β(G)$ is greater than or equal to the zero forcing number $Z(G)$. Our proof of this result is constructive, and yields a polynomial time algorithm to find a zero forcing set with cardinality $β(G)$. We also use the output of \emph{TxGraffiti} to construct several infinite families of claw-free graphs for which $Z(G)=β(G)$. A…
In this paper we extend a result about a sufficient condition for Hamiltonicity for finite graphs by Broersma and Veldmann to locally finite graphs. In order to do this we use topological circles within the Freudenthal compactification of a locally finite graph as infinite cycles. The condition we focus on in this paper is in terms of forbidden induced subgraphs, namely being claw-free and a relaxation of being paw-free.
The maximum oriented $k$-forcing number of a simple graph $G$, written $\MOF_k(G)$, is the maximum directed $k$-forcing number among all orientations of $G$. This invariant was recently introduced by Caro, Davila and Pepper in [CaroDavilaPepper], and in the current paper we study the special case where $G$ is the complete graph with order $n$, denoted $K_n$. While $\MOF_k(G)$ is an invariant for the underlying simple graph $G$, $\MOF_k(K_n)$ can also be interpreted as an interesting property for tournaments. Our main results further focus on the case when $k=1$. These include a lower bound on …
Let $G$ be a simple graph whose vertices are partitioned into two subsets, called filled vertices and empty vertices. A vertex $v$ is said to be forced by a filled vertex $u$ if $v$ is a unique empty neighbor of $u$. If we can fill all the vertices of $G$ by repeatedly filling the forced ones, then we call an initial set of filled vertices a forcing set. We discuss the so-called failed forcing number of a graph, which is the largest cardinality of a set which is not forcing. Answering the recent question of Ansill, Jacob, Penzellna, Saavedra, we prove that this quantity is NP-hard to compute. …
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…
We introduce a family of reductions for removing proper and homogeneous pairs of cliques from a graph G. This family generalizes some routines presented in the literature, mostly in the context of claw-free graphs. These reductions can be embedded in a simple algorithm that in at most |E(G)| steps builds a new graph G' without proper and homogeneous pairs of cliques, and such that G and G' agree on the value of some relevant invariant (or property).
Let each vertex of a graph G = (V(G), E(G)) be given one of two colors, say, "black" and "white". Let Z denote the (initial) set of black vertices of G. The color-change rule converts the color of a vertex from white to black if the white vertex is the only white neighbor of a black vertex. The set Z is said to be a zero forcing set of G if all vertices of G will be turned black after finitely many applications of the color-change rule. The zero forcing number of G is the minimum of |Z| over all zero forcing sets Z \subseteq V (G). Zero forcing parameters have been studied and applied to the m…
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 numbe…
Amos et al. (Discrete Appl. Math. 181 (2015) 1-10) introduced the notion of the $k$-forcing number of graph for a positive integer $k$ as the generalization of the zero forcing number of a graph. The $k$-forcing number of a simple graph $G$, denoted by $F_k(G)$, is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process 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…
Given a finite graph $G$, the maximum length of a sequence $(v_1,\ldots,v_k)$ of vertices in $G$ such that each $v_i$ dominates a vertex that is not dominated by any vertex in $\{v_1,\ldots,v_{i-1}\}$ is called the Grundy domination number, $γ_{\rm gr}(G)$, of $G$. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that $γ_{\rm gr}(G) \geq \frac{n + \lceil \frac{k}{2} \rceil - 2}{k-1}$ holds for every connected $k$-regular graph of order $n$ different from $K_{k+1}$ and $\bar{2…
In this work, we continue the tradition initiated by Geschke, 2011 of viewing the uncountable Borel chromatic number of analytic graphs as cardinal invariants of the continuum. We show that various uncountable Borel chromatic numbers of closed graphs can be consistently different, as well as consistently equal to the continuum. This is done using arguments that are typical to Axiom A forcing notions.
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 rec…
The zero forcing number is a graph invariant introduced to study the minimum rank of the graph. In 2008, Aazami proved the NP-hardness of computing the zero forcing number of a simple undirected graph. We complete this NP-hardness result by showing that the non-equivalent problem of computing the zero forcing number of a directed graph allowing loops is also NP-hard. The rest of the paper is devoted to the strong controllability of a networked system. This kind of controllability takes into account only the structure of the interconnection graph, but not the interconnection strengths along the…
