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K-tree
About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.
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TL;DR: In this work, classes of graphs are found, defined by means of conditions on the clique size and the structure of theClique intersections, whose second iterated clique graphs are also third iteratedClique graphs.
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TL;DR: It will be shown that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four and that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in cases of three colours.
Abstract: A graph G is dually chordal if there is a spanning tree T of G such that any maximal clique of G induces a subtree in T. This paper investigates the Colourability problem on dually chordal graphs. It will show that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in case of three colours. In addition, it will be shown that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four.
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TL;DR: In this paper, the authors prove a necessary and sufficient condition for a clique graph to be complete when $G =G_1+G_2 and G_2 = G. The authors also give a characterization for clique convergence of join of graphs.
Abstract: Let $G$ be a graph and $\mathcal{K}_G$ be the set of all cliques of $G$, then the clique graph of G denoted by $K(G)$ is the graph with vertex set $\mathcal{K}_G$ and two elements $Q_i,Q_j \in \mathcal{K}_G$ form an edge if and only if $Q_i \cap Q_j
eq \emptyset$. Iterated clique graphs are defined by $K^0(G)=G$, and $K^n(G)=K(K^{n-1}(G))$ for $n>0$. In this paper we determine the number of cliques in $K(G)$ when $G=G_1+G_2$, prove a necessary and sufficient condition for a clique graph $K(G)$ to be complete when $G=G_1+G_2$, give a characterization for clique convergence of the join of graphs and if $G_1$, $G_2$ are Clique-Helly graphs different from $K_1$ and $G=G_1 \Box G_2$, then $K^2(G) = G$.
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TL;DR: In this paper, the number of disjoint cliques in the complement graph and the sum of permanent functions over all principal minors of the adjacency matrix of the graph is computed.
Abstract: We give a new recursive method to compute the number of cliques and cycles of a graph. This method is related, respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph. In particular, let $G$ be a graph and let $overline {G}$ be its complement, then given the chromatic polynomial of $overline {G}$, we give a recursive method to compute the number of cliques of $G$. Also given the adjacency matrix $A$ of $G$ we give a recursive method to compute the number of cycles by computing the sum of permanent function of the principal minors of $A$. In both cases we confront to a new computable parameter which is defined as the number of disjoint cliques in $G$.