![]() ![]() The transition with such a random firing time is called an EXP transition. ![]() In particular, random firing times are allowed to obey exponential distributions. SPNs allow random delay times of firing from the time when transitions are enable. In a deterministic PN, transitions immediately fire when the transitions are enable. In the PN modeling, markings provide the state of a target system.Ī stochastic PN (SPN) is defined as the PN which has random firing times. A marking of a PN is given by a vector that represents the number of tokens for all the places. Then the transition is said to be enable. The firing of a transition occurs only when there is at least one token for each input place of the transition. When a transition fires, it removes a token from each input place of the transition, and puts a token to each output place of the transition. Tokens (represented by dots) are located at places in PNs. On the other hand, the place that be connected from a transition is called an output place of the transition. The place that connects to a transition is called an input place of the transition. Directed arcs connect places to transitions and transitions to places. Places and transitions in PNs are represented by circles and rectangles, respectively. Petri net (PN) is a directed bipartite graph with two types of nodes: place and transitions. Dohi, in Mathematics Applied to Engineering, 2017 4.1 MRSPN Hence the result follows from Lemma 7.3.7 and Theorem 7.2.6. Proof.Ĭlearly the graph B′ in Lemma 7.3.7 is a split graph with a split partition ( X, Y). Corollary 7.3.8įor every fixed k ≥ 3, it is NP-complete to determine if a given graph G has threshold dimension at most k, even if G is a split graph. Recall that a graph G is called a split graph with a split partition ( K, S) if V( G) can be partitioned into a clique K and a stable set S. This proves the assertion, and so ch( B) ≤ k. In either case B′ i is not a threshold graph by Theorem 1.2.4, a contradiction. Then the subgraph of B′ i induced by these four vertices is either a P 4 (if it contains the edge xx′) or a 2 K 2 (if it does not). Assume that, if possible, B i has a pair of independent edges xy, x′y′ with x, x′ ∈ X and y, y′ ∈ Y. We assert that they are also difference graphs. For each i, let B i consist of the edges of B′ i between X and Y. ![]() Therefore t( B′) ≤ k.Ĭh( B) ≤ t( B′): Let B′ 1,…, B′ k be threshold subgraphs of B′ that cover its edges. Then by Theorem 2.1.9, the B′ i's are threshold subgraphs of B′ that cover its edges. T( B′) ≤ ch( B): Let B 1,…, B k be difference subgraphs of B that cover its edges, and let B′ 1,…, B′ k respectively be obtained from them by making X into a clique. Let B′ be obtained from B by adding all edges between vertices in X (i.e., making X into a clique). Let B be a bipartite graph whose vertices are partitioned into stable sets X and Y. ![]() Recall that the threshold dimension t( G) of a graph G is the minimum number of threshold subgraphs required to cover E( G). In Annals of Discrete Mathematics, 1995 7.3.4 Threshold Dimension ![]()
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