Wings and perfect graphs

Total perfect codes‎, ‎OO-irredundant and total subdivision in graphs

‎Let $G=(V(G),E(G))$ be a graph‎, ‎$gamma_t(G)$. Let $ooir(G)$ be the total domination and OO-irredundance number of $G$‎, ‎respectively‎. ‎A total dominating set $S$ of $G$ is called a $textit{total perfect code}$ if every vertex in $V(G)$ is adjacent to exactly one vertex of $S$‎. ‎In this paper‎, ‎we show that if $G$ has a total perfect code‎, ‎then $gamma_t(G)=ooir(G)$‎. ‎As a consequence, ...

Perfect $2$-colorings of the Platonic graphs

In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

Perfect Matchings in Edge-Transitive Graphs

We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in G, is the graph constructed from by deleting edges with an en...

Rank-perfect and" weakly rank-perfect graphs

On clique-perfect and K-perfect graphs

A graph G is clique-perfect if the cardinality of a maximum cliqueindependent set of H is equal to the cardinality of a minimum cliquetransversal of H, for every induced subgraph H of G. When equality holds for every clique subgraph of G, the graph is c–clique-perfect. A graph G is K-perfect when its clique graph K(G) is perfect. In this work, relations are described among the classes of perfec...