Graphs with Extremal Connected
Zero forcing is an iterative graph coloring process where at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. The zero forcing number of a graph is the cardinality of the smallest set of initially colored vertices which forces the entire graph to eventually become colored. Connected forcing is a variant of zero forcing in which the initially colored set of vertices induces a connected subgraph; the analogous parameter of interest is the connected forcing number. In this paper, we characterize the graphs with connected forcing numbers 2 and . Our results extend existing characterizations of graphs with zero forcing numbers 2 and ; we use combinatorial and graph theoretic techniques, in contrast to the linear algebraic approach used to obtain the latter. We also present several other structural results about the connected forcing sets of a graph.
Keywords: Connected forcing, zero forcing, separating set, extremal
Zero forcing is an iterative graph coloring process where at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. The zero forcing number of a graph is the cardinality of the smallest set of initially colored vertices which forces the entire graph to be colored. Zero forcing was initially used to bound the maximum nullity of the family of symmetric matrices described by a graph [AIM-Workshop], but has since found a variety of uses in physics, logic circuits, coding theory, power network monitoring, and in modeling the spread of diseases and information in social networks; see [zf_tw, quantum1, logic1, fallat, powerdom3, proptime1, zf_np, powerdom2] for more details.
Connected forcing is a variant of zero forcing in which the initially colored set of vertices induces a connected subgraph. The connected forcing number of a graph is the cardinality of the smallest connected set of initially colored vertices which forces the entire graph to be colored. Various structural and computational aspects of connected forcing have been investigated in [brimkov, brimkov2, CF_paper]. The connected forcing number bounds parameters such as the maximum nullity, path cover number, and power domination number (cf. [brimkov, CF_paper]); it can also potentially be applied to power network monitoring and modeling propagation of information (cf. [brimkov2]). Other variants of zero forcing, such as positive semidefinite zero forcing [Barioli, positive_zf2, proptime2], fractional zero forcing [fractional_zf], and signed zero forcing [signed_zf] have also been studied; see also [butler, kenter] and the bibliographies therein. These variants are typically obtained by modifying the zero forcing color change rule, or adding certain restrictions to the structure of a forcing set.
Computing the zero forcing number and connected forcing number of a graph are both NP-complete problems [aazami, brimkov2]; thus, approaches addressing the complexity of these problems include developing closed formulas, characterizations, and bounds for the forcing numbers of graphs with special structure; such results are obtained in [AIM-Workshop, benson, brimkov, brimkov2, Eroh, Huang, Meyer]. In particular, graphs whose zero forcing number equals , , and have been characterized in [row], and graphs whose zero forcing number equals have been characterized in [AIM-Workshop]; similarly, graphs whose connected forcing number equals 1 and have been characterized in [brimkov]. In this paper, we extend these results by characterizing graphs whose connected forcing numbers are 2 and . Other related characterizations have been derived for graphs whose minimal rank is two [barrett, barrett1] and three [barrett2], graphs whose positive semi-definite matrices have nullity at most two [holst1], three-connected graphs whose maximum nullity is at most three [holst2], and graphs for which the maximum multiplicity of an eigenvalue is two [johnson]. Many of these characterizations have been obtained using linear algebraic approaches; in contrast, we employ novel combinatorial and graph theoretic techniques which make use of the vertex connectivity of a graph and the connectedness of its forcing set. We also present several other structural results, and introduce a generalization of zero forcing whose further study could be of independent interest.
The paper is organized as follows. In the next section, we recall some graph theoretic notions, specifically those related to zero forcing. In Section 3, we characterize graphs with connected forcing numbers 2 and , and present several other structural results about connected forcing sets. We conclude with some final remarks and open questions in Section 4.
2.1 Graph theoretic notions
A graph consists of a vertex set and an edge set of two-element subsets of . The order and size of are denoted by and , respectively. Two vertices are adjacent, or neighbors, if . If is adjacent to , we write ; otherwise, we write . The degree of a vertex in , denoted , is the number of neighbors has in ; the dependence on can be ommitted when it is clear from the context. The minimum degree and maximum degree of are denoted by and , respectively. A leaf, or pendant, is a vertex with degree 1. An isolated vertex or isolate is a vertex with degree 0; such a vertex will also be called a trivial (connected) component of . Given , the induced subgraph is the subgraph of whose vertex set is and whose edge set consists of all edges of which have both endpoints in . An isomorphism between graphs and will be denoted by . The number of connected components of a graph will be denoted by .
A separating set of is a set of vertices which, when removed, increases the number of connected components in . A cut vertex is a separating set of size one. The vertex connectivity of , denoted , is the largest number such that remains connected whenever fewer than vertices of are removed; a disconnected graph has vertex connectivity zero. A cut edge is an edge which, when removed, increases the number of connected components of . A biconnected component, or block, of is a maximal subgraph of which has no cut vertices; is biconnected if it has no cut vertices. An outer block is a block which contains at most one cut vertex of . A trivial block is a block with two vertices, i.e., a cut edge of .
The disjoint union of sets and , denoted , is a union operation that indexes the elements of the union set according to which set they originated in; the disjoint union of two graphs and , denoted , is the graph . The join of two graphs and , denoted , is the graph obtained from by adding an edge from each vertex of to each vertex of . The complement of a graph is the graph . A complete graph is denoted , and a complete bipartite graph, denoted is the complement of (we may allow these indices to equal 0, in which case ). A graph with no edges will be called an empty graph; a path on vertices will be denoted . If is a set of graphs, a graph is -free if it does not contain as an induced subgraph for each . For other graph theoretic terminology and definitions, we refer the reader to [bondy].
2.2 Zero forcing
Given a graph and a set of initially colored vertices, the color change rule dictates that at each integer-valued time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored; such a force is denoted . The derived set of is the set of colored vertices obtained after the color change rule is applied until no new vertex can be forced; it can be shown that the derived set of is uniquely determined by (see [AIM-Workshop]). A zero forcing set is a set whose derived set is all of ; the zero forcing number of , denoted , is the minimum cardinality of a zero forcing set.
A chronological list of forces of is a sequence of forces applied to obtain the derived set of in the order they are applied; there can also be initially colored vertices which do not force any vertex. Generally, the chronological list of forces is not uniquely determined by ; for example, it may be possible for several colored vertices to force an uncolored vertex at a given step. A forcing chain for a chronological list of forces is a maximal sequence of vertices such that for . It may be possible for an initially colored vertex not to force any vertex. If a vertex forces another vertex at some step of the forcing process, then it cannot force a second vertex at a later step, since that would imply it had two uncolored neighbors when it forced for the first time. Thus, each forcing chain induces a distinct path in , one of whose endpoints is an initially colored vertex and the rest of whose vertices are uncolored at the initial time step; we will say the initially colored vertex initiates the forcing chain. The set of all forcing chains for a chronological list of forces is uniquely determined by the chronological list of forces and forms a path cover of .
A connected zero forcing set of is a zero forcing set of which induces a connected subgraph. The connected zero forcing number of , denoted , is the cardinality of a minimum connected zero forcing set of . For short, we may refer to these as connected forcing set and connected forcing number. Note that a disconnected graph cannot have a connected forcing set.
3 Graphs with extremal connected forcing numbers
Polynomial time algorithms and closed-form expressions have been derived for computing the connected forcing numbers of special classes of graphs, including trees, unicyclic graphs, grid graphs, sun graphs, and several other families (cf. [brimkov, brimkov2]). Conversely, a complete characterization of graphs having a particular connected forcing number can be obtained through a combinatorial case analysis. For example, it is easy to see that if and only if is a path . Moreover, Brimkov and Davila [brimkov] gave the following characterization of graphs with connected forcing number .
[brimkov] if and only if , , or , .
In what follows, we extend these results by characterizing graphs for which and .
3.1 Graphs with
In this section, we will characterize all graphs with connected forcing number 2. We first recall some definitions and previous results.
A pendant path attached to vertex in graph is a set such that is a connected component of which is a path, one of whose ends is adjacent to in . The neighbor of in will be called the base of the path, and will denote the number of pendant paths attached to .
Let be a connected graph. Define
When there is no scope for confusion, the dependence on will be omitted.
[brimkov2] Let be a connected graph different from a path and be an arbitrary connected forcing set of . Then .
A graph is a graph of two parallel paths specified by and if , and if can be partitioned into nonempty sets and such that and are paths, and such that can be drawn in the plane in such a way that and are parallel line segments, and the edges between and (drawn as straight line segments) do not cross; such a drawing of is called a standard drawing. In a standard drawing of , fix an ordering of the vertices of and that is increasing in the same direction for both paths. In this ordering, let and respectively denote the first and last vertices of for . The sets and will be referred to as ends of .
Note that if is a graph of two parallel paths, there may be several different partitions of into and which satisfy the conditions above. For example, let be a cycle on 5 vertices. Then is a graph of two parallel paths that can be specified by and , as well as by and .
Graphs of two parallel paths were introduced by Johnson et al. [johnson] in relation to graphs with maximum nullity 2. They were also used by Row [row] in the following characterization.
[row] if and only if is a graph of two parallel paths.
The following observation regarding the result of Theorem 3.2 is readily verifiable (and has been noted in [row]).
Either end of a graph on two parallel paths is a zero forcing set. Conversely, if , the two forcing chains associated with a minimum zero forcing set induce a specification of as a graph on two parallel paths.
The following observation follows from the definition of forcing vertices.
Every minimum zero forcing set and every minimum connected forcing set contains a vertex together with all-but-one of its neighbors.
Finally, let denote the set of leaves of ; we recall a result of Brimkov and Davila [brimkov] relating to .
[brimkov] For any connected graph different from a path, .
We now prove the main result of this section.
if and only if belongs to the family of graphs described in Figures LABEL:fig_cf2case1 and LABEL:fig_cf2case2.
Let be a graph with . Since if and only if , and since , it follows that . Thus, by Theorem 3.2, is a graph of two parallel paths. Fix some partition of into and which satisfies Definition 3, fix a standard drawing of based on that partition, and fix a vertex ordering as specified in Definition 3. From Lemma 2, it follows that has 0, 1, or 2 leaves.
Let be a graph of two parallel paths with . Then, there are at least two edges between the two parallel paths of .